HOW COMPETENT MATHEMATICS TEACHERS DEVELOP PEDAGOGICAL CONTENT KNOWLEDGE IN STATISTICS TEACHING BY

HOW COMPETENT MATHEMATICS TEACHERS DEVELOP PEDAGOGICAL CONTENT KNOWLEDGE IN STATISTICS TEACHING BY
HOW COMPETENT MATHEMATICS TEACHERS DEVELOP
PEDAGOGICAL CONTENT KNOWLEDGE IN STATISTICS
TEACHING
BY
SUNDAY BOMBOI IJEH
Submitted in partial fulfilment of the requirement for the degree
PhD (MATHEMATICS EDUCATION)
Department of Science, Mathematics and Technology Education
Faculty of Education
University of Pretoria
Pretoria
Supervisor: Professor G.O.M. Onwu
AUGUST, 2012
i © University of Pretoria
CERTIFICATION
This thesis has been examined and approved as meeting the required
standard of scholarship for the fulfilment of the Degree of Doctor of
Philosophy in Mathematics Education.
Prof. G. O. M. Onwu
............................................
Date................................
SUPERVISOR
ii UNIVERSITY OF PRETORIA
DECLARATION OF ORIGINALITY
Full names of student ……SUNDAY BOMBOI IJEH
Students number ……27488111
Declaration
1. I understand what plagiarism is and am aware of the university’s policy in this regard.
2. I declare that this THESIS is my own original work. Where other peoples’ work has
been used, this has been properly acknowledged and referenced in accordance with
departmental requirements.
3. I have not used work previously produced by another student or any other person to
hand in as my own.
4. I have not allowed, and will not allow, anyone to copy my work with the intention of
passing it off as his or her own work.
Signature of student.........................................................
Signature of Supervisor……………………………………..
iii ACKNOWLEDGEMENTS
I would like to express my heartfelt gratitude to the following people for their contribution
towards the successful completion of this project.
•
The Almighty God for giving me the strength, wisdom and courage to complete this
project. Without Him, I would not have made it.
•
My supervisor: Professor Gilbert Oke Onwu for his patience, invaluable guidance and
encouragement throughout this research project. It was indeed a privilege to have
worked with him. His constructive criticism and quick responses led to the successful
completion of this project.
•
To Professor Irma Eloff, Professor W. M. Braun, Professor K. Dvzimbo, Professor L.
D. M. Lebeloane and Dr E. C. Ochonogor for their moral and courageous support.
•
To Professor L.C. Jita, Professor D. Mogari, Professor L. Nyaumwe, Dr M. L.
Mokhele, Dr A. Motlhabane and Mr M. Phoshoko for their professional advice and
mentorship throughout the study.
•
To Mr James Matshego for helping to lay the foundation for this study.
•
The Gauteng Department of Education, for permission to conduct the research in
public schools.
•
To my wife and family for encouragement, patience and support.
iv TABLE OF CONTENTS
CHAPTER 1
Introduction
1.1
Background to the study
1
1.2
The research problem
5
1.3
Aims of the Study
6
1.4
Statement of the problem
6
1.5
Research questions
6
1.6
Significance of the study
7
1.7
Theoretical framework
8
1.7.1 Subject matter content knowledge
11
1.7.2 Pedagogical knowledge
12
1.7.3 Knowledge of learner’s conception in statistics teaching in school
13
mathematics
1.7.4 Knowledge of Learners’ learning difficulties
14
1.8
Scope of the study
15
1.9
Criteria for selecting the topic
15
1.10
Definition of terms
16
1.11
The chapter structure of the thesis
18
1.12
Summary of the chapter
20
v CHAPTER 2
2.0 LITERATURE REVIEW
2.1
Introduction
21
2.2
National Curriculum Statement for mathematics and statistics
21
2.3
Research on teaching of statistics in school mathematics
23
2.4
Assessing Teachers’ PCK
25
2.4.1
Description of PCK
25
2.4.2
Teacher knowledge and PCK
29
2.4.3
Pedagogical content knowledge and subject matter for teaching
31
2.4.4
PCK and pedagogical knowledge (Instructional skills and
strategies)
38
2.4.5
PCK and knowledge of learners’ preconceptions and learning
difficulties
42
Summary of the chapter
44
2.6
CHAPTER 3
3.0 RESEARCH METHODOLOGY AND PROCEDURE
3.1
Introduction
46
3.2
Assumption of PCK development during classroom
practice
Research design and methods used in this study
46
46
3.3.1
Research design
46
3.2.2
Research methods
46
Population and sample description
47
3.4.1
Study population
47
3.4.2
Study sample
48
3.3
3.4
vi 3.5
3.5.1
Research instruments used for collecting data
49
Development of instrument
49
3.5.1.1 Teacher conceptual knowledge exercise in statistics
49
3.5.1.2 Concept mapping for teachers
52
3.5.1.3 Interview schedule for teachers
53
3.5.1.4 Lesson observation schedule
55
3.5.1.5 The teacher questionnaire
55
3.5.1.6 Teacher written reports
57
3.5.17
58
Document analysis
3.5.1.8 Video recording
3.6
Validation of the research instrument
59
3.6.1
Validity and reliability of the concept map
59
3.6.2
Validity and reliability of the interview schedule 60
The Pilot study
61
3.7.1
Subjects used in the pilot study
61
3.7.2
Administration of pilot study
61
3.7.3
Result of the pilot study
62
3.7.3.1
The conceptual knowledge exercise
62
3.7.3.2
The concept mapping
68
3.7.3.3
The lesson observation schedule
68
3.7.3.4
The interview schedule
69
3.7.3.5
The questionnaire for teachers
69
3.7.3.6
The written report guide
70
The Main study
70
3.8.1
Subjects used in the Main study
70
3.8.2
Administration of the Main study
70
3.7
3.8
vii 58
3.9
Data analysis and results of the Main study
71
3.9.1
Quantitative data analysis
71
3.9.2
Qualitative data analysis
71
3.10
Ethical issues
72
3.11
Summary of the chapter
72
CHAPTER 4
4.0 DATA ANALYSIS AND DISCUSSION
4.1
Introduction
73
4.2
Conceptual knowledge exercise
73
4.3
Teacher demographic profile
73
4.4
Concept mapping
74
4.5
Classroom practice (Lesson observation)
74
4.5.1
Lesson observation of teacher A
75
4.5.2
Lesson observation of teacher B
96
4.5.3
Lesson observation of teacher C
110
4.5.4
Lesson observation of teacher D
127
Video Recording of the lesson observation of the four
teachers
141
4.6
4.7
Teacher development of PCK
141
4.7.1
Teacher development of subjects matter content knowledge
141
4.7.2
Teacher development of pedagogical knowledge (instructional
143
skill and strategies)
4.7.3
Teacher development of knowledge of learner’s preconception
viii 146
and learning difficulties
4.7.4
Teacher development of PCK in statistics teaching
4.8
Summary of the chapter
149
149
CHAPTER 5
5.0 Discussion of results
5.1
Introduction
151
5.2
Teacher development of PCK
152
5.2.1 Teacher A
152
5.2.2 Teacher B
159
5.2.3 Teacher C
165
5.2.4 Teacher D
170
5.3
Evaluation of theoretical framework
175
5.4
Summary of chapter
179
CHAPTER 6
6.0 Summary and recommendations of the study
6.1
Introduction
180
6.2.
Focus of the study
180
6.3
Summary of the results according to the theoretical
framework
182
6.3.1
Subject matter content knowledge
182
6.3.2
Pedagogical knowledge (instructional skills and strategies)
182
6.3.3
Knowledge of learners’ preconceptions and learning
difficulties
184
ix 6.4
Concluding remarks
185
6.5
Educational implication of the study
186
6.6
Suggestions for further study
187
6.7
Limitations of the Study
188
6.8
The role of the researcher in the non-participatory lesson 189
observation.
6.9
Summary of the chapter 190
BIBLOGRAPHY
192
APPENDICES
206
x ABSTRACT
This study is concerned with how competent mathematics teachers develop pedagogical
content knowledge (PCK) in statistics teaching. Pedagogical content knowledge was used as
the theoretical framework that guided the research and data collection.
The study’s methodology consisted of two phases. In the first phase, the six identified
mathematics teachers undertook a conceptual knowledge written exercise. The result of this
exercise was used to select the best four performing teachers for the second phase of the
study. The second phase consisted mainly of lesson observations, interviews, written
documents in the form of completed questionnaires, written diaries or reports, document
analysis designed to produce rich detailed descriptions of participating teachers’ PCK in the
context of teaching statistics concepts at school level. The concept mapping exercise was
used to indirectly assess participating teachers’ content knowledge and their conceptions of
the nature of school statistics and how it is to be taught. The qualitative data obtained were
analysed to try to determine individual teachers’ content knowledge of school statistics,
related pedagogical knowledge, knowledge of learners’ conceptions in statistics teaching,
knowledge of learners’ learning difficulties as well as how they developed their PCK in
statistics teaching. The analysis was done based on iterative coding and categorisation of
responses and observations made to identify themes, patterns, and gaps, in school statistics
teaching. Commonalities and differences if any, in the PCK profiles of the four participating
teachers were also analysed and determined.
The results of the study showed that overall, individual teachers develop their PCK in school
statistics teaching by:
(a) formally developing their knowledge of the subject matter in a formal undergraduate
educational programme, as well as subject matter content knowledge during classroom
practice;
(b) using varied topic-specific instructional skills such as graphical construction skills in
teaching statistical graphs;
xi (c) using diagnostic techniques (oral questioning and pre-activity, class discussions and
questioning) and a review of previous lessons to introduce lessons, and to determine
learners’ preconceptions in statistics teaching ;
(d) Using teaching strategies that can help to identify learners’ learning difficulties as well
as intervention to address the difficulties;
(e) continually updating their knowledge of school statistics by attending content knowledge
workshops and other teacher development programmes designed to improve content
knowledge and practice.
Keywords: pedagogical content knowledge (PCK), subject matter content knowledge,
pedagogical knowledge, instructional strategies, conceptions, learning difficulties, competent
teachers, data handling, procedural knowledge, conceptual knowledge.
xii LIST OF TABLES
Table 1.1
Learners’ performance in mathematics from 1999 to 2004 in the 3
South African Senior Certificate examination.
Table 3.1
Schools and teachers that participated in the main study
48
Table 3.2
Mathematics assessment taxonomy and marks allocation
49
Table 3.3
Showing competency and skills and marks allocated
50
Table 3.4
A table showing the list of statistics taught in grades 10, 11 and 12 52
(if any).
Table 3.5
Item specification table for the interview
54
Table 3.6
Item specification table for the questionnaire
56
Table 3.7
Item specification table for the written reports
57
Table 3.8
Item response pattern of the conceptual knowledge exercise from 64
the pilot study test items
Table 3.9
Summary of discrimination indices of the test items
66
Table 3.10
Summary of difficulty indices of the test items
67
Table 3.11
Summary of test characteristics
68
Table 4.2
Teacher A, B, C, and D profiles
74
Table 4.5.1
Description of classroom condition and lesson observation of teacher A
76
Table 4.5.1a
A frequency table of members of netball club
79
Table 4.5.1b
A frequency table showing learners’ performance in a test
85
Table 4.5.2a
Description of lesson observation and classroom condition at school 97
xiii B
Table 4.5.2b:
A frequency table showing the performance of learners in a test
Table 4.5.2c
A frequency table showing the distribution of the amount spent on buying 101
98
some groceries from a supermarket
Table 4.5.2d
Mark distribution of learners in an English examination
104
Table 4.5.3a
Description of lesson observation and classroom condition in Teacher C’s
111
mathematics lesson
Table 4.5.3b:
A table showing the ages of cars in a sample of 100 car owners
Table 4.5.3c
A frequency table showing the distribution of learners’ performance 123
113
in two tests
Table 4.5.4a
Description of lesson observation and classroom condition at school 128
D
Table 4.5.4bi
A table showing the number of different makes of cars in a car park
128
Table 4.5.4bii
A frequency table showing the mark distribution of learners in a class test
131
Table 4.5.4c
A frequency table showing the masses of players in a 2003 South African
135
rugby squad
Table 4.7.1
Teachers’ responses to interview about teachers’ subject matter
content 242
knowledge in statistics teaching
Table 4.7.2
Participants’ responses to the interview, questionnaire and written reports about
245
teachers’ knowledge of instructional skills and strategies for teaching statistics.
Table 4.7.3
Participants responses to the questionnaire and written reports on teachers’ 250
knowledge of learners’ preconceptions and misconceptions in statistics
teaching
Table 4.7.4
Participants responses to the teachers’ interview, questionnaire and written 252
reports about teachers’ knowledge of learners’ learning difficulties
Table 4.9
A comparison of the documents used by participants in statistics teaching
xiv 255
LIST OF FIGURES
Figure 2.1
The components of PCK used in this study
31
Figure 4.5.1a
A histogram showing the age distribution of members of a netball club
81
Figure 4.5.1b
A histogram showing the age distribution of members of a netball club
82
Figure 4.5.1c
An example of an incomplete classwork exercise on a histogram,
86
showing the mark distribution of learners’ performance in a test
Figure 4.5.1d
A box-and-whisker plot showing the marks obtained in an examination.
89
Figure 4.5.1e
An ogive showing the mark distribution of learners in an English
90
examination
Figure 4.5.2a
A bar graph of the scores of learners in a test used to explain how to 99
construct, analyse, and interpret a bar graph using the scores in line
Figure 4.5.2b
An ogive representing learner performance in an English examination
105
Figure 3.5.3a
Ogive of age distribution of sample of 100 cars owners park in a car park
114
Figure 4.5.3b
Scatter diagrams showing the relationships between X and Y
121
Figure 4.5.4a
A bar graph showing the number of different makes of cars in a car park
130
Figure 4.5.4b
A histogram showing the distribution of the masses of players in a 2003 136
South African rugby squad
xv LIST OF APPENDICES
Appendix I
Appendix II
C Consent letter for participating teachers
Request for permission to allow your child to participate
206
208
in a research programme in mathematics
Appendix IIIA
Request for permission to conduct research on how competent
209
mathematics teachers develop pedagogical content knowledge in
statistics teaching in your schools
Appendix IIIB
Approval letter to conduct research in high schools in Gauteng
211
Province
Appendix IV
Criteria for validating interview schedule for teacher on
213
how they develop PCK in statistics teaching.
Appendix V
Transcription of video records of first lesson observation of
216
teacher A
Appendix VI
Transcription of video records of second lesson observation
219
of teacher A
Appendix VII
Transcription of video records of first lesson observation of
221
teacher B
Appendix VIII
Transcription of video records of second lesson observation of
223
teacher B
Appendix IX
Transcription of video records of first lesson observation of
225
teacher C
Appendix X
Transcription of video records of second lesson observation of
228
teacher C
Appendix XI
Transcription of video records of first lesson observation of
230
teacher D
Appendix XII
Transcription of video records of second lesson observation of
teacher D
xvi 232
APPENDIX XIII
Criteria for validating questionnaire schedule for teacher on how
234
they develop PCK in statistics teaching.
Appendix XIV
Criteria for validating written reports schedule for teacher on
236
how they develop PCK in statistics teaching.
Appendix XV
Criteria for validating document analysis schedule for teachers
237
on how they develop PCK in statistics teaching.
Appendix XVI
Appendix XVII
Criteria for validating the lesson plan and observation schedule. 240
Analysis of participants’ responses to interview, questionnaire
242
and teachers’ written report
Appendix XVIII
Participants’ responses to the interview, questionnaire and
245
written reports about teachers’ knowledge of instructional skills
and strategies for teaching statistics.
Appendix XIX
Teachers’ knowledge of learners’ preconceptions and
250
misconceptions.
Appendix XX
Knowledge of learners’ learning difficulties
252
Appendix XXI
Comparison of the texts used by participants in statistics
255
teaching
Appendix XXII
An exercise in statistics for mathematics teachers
258
Appendix XXIII
Memo for final conceptual knowledge exercise, march 2010
267
Examining the content knowledge of mathematics teachers in
268
Appendix XXIV
statistics teaching
Appendix XXV
Rubric for concept mapping exercise
270
Appendix XXVI
The interview schedule for mathematics teachers
273
Appendix XXVII
Report on the teaching of statistics
278
Appendix XXVIII
The questionnaire for mathematics teachers
280
Appendix XXIX
Instrument validation form
284
Appendix XXX
Ethical clearance certificate
286
xvii Appendix XXXI
A sample of teachers’ response to concept mapping exercise
286
Appendix XXXII
Lesson observation sheet
288
xviii CHAPTER ONE
1.0
INTRODUCTION
1.1
Background to the study
This study focused on how competent secondary school mathematics teachers develop
pedagogical content knowledge (PCK) for teaching statistics in high school. While some
researchers perceive statistics as a subject on its own (Moore & Cobb, 2001; Gordon, Petocz
& Reid, 2007), others believe it should be taught as part of the mathematics curriculum and
consequently view it as a mathematical concept (Franklin, Kader, Mewborn, Moreno, Peck,
Perry & Schaeffer, 2005; Gattuso, 2006).
According to the National Curriculum Statements (NCS) of South Africa (DoE, 2009), the
country in which this study was conducted, statistics is taught as part of the mathematics
curriculum under the rubric of ‘data handling’. In accordance with the new curriculum, the
learning outcomes of mathematics require that learners should be able to use appropriate
measures of central tendency and spread to collect, organise, analyse, and interpret data in
order to establish statistical and probability models for solving related problems (DoE, 2007).
According to the NCS, instructional guides and other publications, teachers need to be given
in-service professional support by mathematics experts or professionals with the statistics
knowledge required to implement the new mathematics curriculum. This is because the topic
of statistics has been included in the national curriculum for the first time, and it is assumed
that most teachers will not have the requisite knowledge for teaching it. Thompson (2005)
indicated that in order to implement the new curriculum effectively, teachers need subject
matter knowledge, pedagogical knowledge, and pedagogical content knowledge (PCK).
Subject matter knowledge refers to the disciplinary knowledge obtained through formal
training in colleges and universities, while pedagogical knowledge pertains to the knowledge
of instruction and learning that the teacher needs in order to deal with everyday classroom
educational tasks (Vistro-Yu, 2003). Such tasks involve the use of various teaching styles and
strategies and the management of learning processes in the classroom (Vistro-Yu, 2003).
These skills and competencies are normally acquired through formal training and teaching
practice. Simply described, PCK is about the overall knowledge the educator has of the
subject matter content that learners should master in a particular topic or subject, and how it
1
should be taught, so that effective and efficient learning can take place (Mitchell & Mueller,
2006). In short, PCK is an amalgam of subject matter content and pedagogy, which is
uniquely the province of teachers and involves their own special form of professional
understanding for good teaching (Jong, 2003).
PCK is specific to teaching and differentiates between expert teachers in a particular subject
area and subject area experts (Griffin, Dodds & Rovengno, 1996). To illustrate, mathematics
teachers differ from mathematicians, not necessarily in the quantity and quality of their
subject matter knowledge, but more specifically in how that knowledge is organised and used
(Cochram, De Ruiter & King, 1993). An experienced mathematics teacher’s knowledge of
the subject is organised from a teaching perspective and is used as a basis for helping learners
to understand specific concepts. A mathematician’s knowledge, on the other hand, is
normally organised from a research perspective and is used mainly as a basis for developing
new knowledge in the field. This implies that PCK may be something beginner or
inexperienced teachers may not necessarily learn only from textbooks or from short courses.
From the literature reviewed, little is known as to how PCK is developed, or even facilitated,
in the context of teaching statistics (Godino, Batenero, Roa & Wilhelmi, 2011; DoE 2008;
Jong, 2003). Therefore, further research is needed in order to identify and define the skills
and practices necessary for PCK development in statistics education more carefully (DoE,
2008).
To develop PCK, Jong (2003) argues that teachers need to explore instructional strategies for
specific topics and their learners in practice. Various studies – such as those by Dooren,
Verschattel and Oghenna (2005), Boerst (2003), Halim and Meerah (2002) and Van Driel,
Verloop and De Vos (1998) have shown that inadequate PCK is one of the areas that require
most attention in teacher education, as many teachers are unable to enhance learner
performance because of lack of subject matter content knowledge and PCK. Many beginner
teachers, including inexperienced mathematics teachers, do not know how to develop and use
PCK in their teaching (Van Driel et al., 1998; Halim & Meerah, 2002). In consequence, they
become uncomfortable with teaching certain topics, and, for that reason, may omit them
altogether (ICM/IASE, 2007).
Data on mathematics enrolment and learner performance over a period of five years in the
South African Senior Certificate (SC) examination, as displayed in Table 1.1 and Figure 1.1
2
below, show that learners generally underachieve in mathematics. Mathematics failure rates
in the SC examination remain unacceptably high, and the number or percentage of learners
that leave Grade 12 with a higher-grade pass in mathematics is unacceptably low. While the
percentage of candidates that wrote the mathematics examination over the period of six years
increased, the percentage of learners that passed mathematics for standard grade (MSG) was
below 30%, and below 10% in mathematics for higher grade (MHG) (Figure 1.1). This
suggests a crisis of mathematics underachievement at secondary school level.
Table 1.1:
No. of candidates
% of learners
that wrote
mathematics
% of learners
that passed
mathematics
Learners’ performance in mathematics from 1999 to 2004 in the South
African Senior Certificate Examination
1999
2000
2001
2002
2004
511,225
489,941
449,371
442,590
467,985
55%
58%
59%
59%
81%
SG
HG
SG
HG
SG
HG
SG
HG
SG
HG
20%
4%
21%
5%
24%
4%
27%
5%
29%
5%
Source: DoE (2006); CDE (2007)
No of NB: This is the period in which the standard and higher grades examinations are used
to assess mathematics learners in the Senior Certificate Examination.
3
In addition, the chief examiner’s report on learners’ performance in both mathematics and
mathematical literacy in the 2008 and 2009 SC examinations shows that learners generally
underperform in statistics (DoE, 2009). According to this report, there was a steady increase
in learners’ enrolment and performance in mathematics, compared with previous years, but
about 60% of those who passed scored between 30% and 40% (the pass mark for
mathematics, according to the NCS, is 30%) (DoE, 2009). Furthermore, the learners’
performance in questions relating to statistics in paper 2 was below 35%. As a result of the
poor performance in statistics, teachers’ ability to teach this topic, the quality of senior
certificate products, and university enrolment in mathematics and statistics-related subjects
have been subject to review (Keeton, 2009).
Many studies, such as those by Howie (2002) and DoBE (2012), on the causes of poor
performance in mathematics in South Africa, show that one of the main factors that is
attributable to learners’ poor performance is the teacher. Others include language and
classroom environment (CDE, 2004). The interest in this study is with the teacher factor. The
study is aimed at investigating specifically how competent teachers develop and use PCK to
improve the quality of instruction and learning in statistics. The competent mathematics
teachers were identified from their learners’ final results in mathematics in the public senior
certificate exam and on recommendations by principals, peer teachers and subject experts in
the Department of Education. Although being competent may not necessarily mean that they
are expert in statistics, their selection as competent teachers depends on their final Senior
Certificate Examination results in mathematics over time. The research seeks to determine
what it is that these teachers who have been classified as competent teachers have and do
when using their PCK to teach particular subject matter content in statistics. The assumption
here is that PCK can be measured. PCK has been used as a theoretical framework for this
study.
The topic of statistics has been chosen because it is completely new in the mathematics
curriculum, and many teachers may not have adequate experience in teaching it, let alone in
handling the difficulties learners experience with it. Until the introduction of the topic of data
handling in mathematics and mathematical literacy in 2006, statistics was not taught in high
schools (DoE, 2006). Many, if not all, teachers of mathematics would not have formal
knowledge of statistics, let alone knowledge of learners’ preconceptions, which need to be
4
addressed in teaching and learning statistics. The assumption is that few in-service teachers
would have developed the PCK needed to teach the topic effectively. Therefore, it would be
useful to study how teachers who are considered competent go about teaching a new topic in
statistics, and to document what it is that they have and do as they go about preparing their
lessons, and how they teach those data-handling lessons.
In 2007, this lack of familiarity with statistics content and teaching on the part of secondary
school mathematics teachers worldwide was given added support by papers presented at the
joint conference of the International Commission for Mathematics Instruction and the
International Association for Statistics Educators (ICMI/IASE, 2007). The conference
highlighted that mathematics teachers are likely to face challenges in terms of teaching a
topic such as statistics in which they do not necessarily have an understanding of learners’
learning difficulties, and may not know how to present the content in a way that learners can
understand.
Recent studies, such as those by Jong (2003), Jong, Van Driel and Verloop (2005), Capraro,
Capraro, Parker, Kulm and Raulerson (2005), Wu (2005), and Godino et al. (2011), showed
that most mathematics teachers at high-school level have limited PCK. A clear understanding
of how teachers develop PCK and use it to enhance learner achievement in mathematics is
useful knowledge for any pre-service and in-service teacher education programme. This study
is an attempt to provide a comprehensive description and analysis of how the mathematics
teachers selected for the study developed their PCK in teaching statistics.
1.2
The research problem
The NCS Curriculum for Mathematics was introduced in Grade 10 in all high schools in the
Republic of South Africa in 2006. Mathematics teachers were charged with the responsibility
of delivering the curriculum in the classroom in line with the NCS recommendations and
ensuring effective teaching, so that learner achievement could be enhanced (DoE, 2006).
However, since the introduction of this curriculum, learners have not been performing as they
should, because of internal and external classroom factors that result in underachievement
(Howie, 2002; CDE, 2004; DoE, 2008).
Reddy (2006) identifies PCK as one of the limiting factors in enhancing learner achievement
5
in mathematics in the South African context. Other researchers elsewhere in the world, such
as Wu (2005), Capraro et al. (2005), Halim and Meerah (2002), and Van Driel et al. (1998),
have come to the same conclusion, especially with regard to statistics (Cazorla, 2006), which
has only recently been included in the curriculum as a formal aspect of mathematics. The lack
of familiarity with statistics has placed teachers’ confidence in their ability to teach it in
doubt (ICMI/IASE, 2007). Poor learner performance in statistics was also noted at the joint
conference of the ICMI and the IASE (ICMI/IASE, 2007), at which conference delegates
attributed learners’ poor performance to the rudimentary state of mathematics teachers’ PCK
in statistics. In addition, the chief examiner’s report for the Senior Certificate Examination in
Mathematics shows that learners underperform in statistics (DoE, 2008). The report suggests
that poor PCK background may have contributed to learners’ underperformance in statistics,
and that this background therefore needs to be investigated (DoE, 2010).
Given the instructional demands of the new mathematics curriculum and the poor
performance of learners in statistics, this study was concerned with investigating how
competent mathematics teachers at high-school level in South Africa develop PCK in
statistics teaching in order to enhance learners’ achievement in mathematics.
1.3
Aims of the study
The aims of the study were:
a)
To determine how competent secondary school mathematics teachers develop PCK for
teaching statistics
b)
1.4
To determine the implications that PCK has for mathematics education programmes
Statement of the problem
The problem identified for this study was to determine how secondary school mathematics
teachers who are assumed to be competent develop the PCK they use in teaching statistics in
school mathematics. In addition, the implications of these findings for mathematics teacher
education programmes were determined and discussed.
6
1.5
Research questions
The problem statement gave rise to the following research questions:
1)
What subject matter content knowledge of statistics do mathematics teachers who are
considered competent have and demonstrate during classroom practice?
2)
What instructional skills and strategies do these teachers use in teaching statistics?
3)
What knowledge of learners’ preconceptions and learning difficulties, if any, do these
teachers have and demonstrate during classroom practice?
(4)
1.6
How do these teachers develop PCK in statistics teaching?
Significance of the study
The significance of this study is that it is hoped its findings will provide a knowledge base
and process employed by mathematics teachers to develop pedagogical content knowledge in
statistics teaching for the improvement of learners’ performance; and ideas and knowledge
that can be incorporated into a mathematics education programme for in-service and preservice mathematics teachers.
Besides, PCK development is a complex process and it is not clear how it is developed in
statistics teaching for mathematics classroom practices. ‘PCK is distinct from a general
knowledge of pedagogy, educational purpose and learners’ characteristics’ (Jong, Van Driel
& Verloop, 2005: 948). ‘Moreover, because PCK is concerned with the teaching of a
particular topic for example statistics, it may turn out to differ considerably from the subject
matter itself’ (Jong, Van Driel & Verloop, 2005: 948). PCK is said to develop by an iterative
process that is rooted in classroom practice (Miller, 2006). The implication is that beginning
teachers have little or no PCK at their disposal, particularly if they are new to statistics
teaching. A clear understanding of how PCK is developed in statistics teaching will be a
requisite for designing effective statistics education programme for in-service and pre-service
statistics educators.
A great deal of research has been conducted in an attempt to identify and characterise PCK
during classroom practice, but research communities continue to call for studies to devise
methods of measuring PCK (Miller, 2006). According to Miller (2006), PCK represents
much more than a category of teacher knowledge; it provides a starting point for research
7
involving teacher education. As a theoretical framework of this study, PCK provides a
process for organising teacher education research.
1.7
Theoretical framework
Several researchers (Shulman, 1986; Van Driel, 1998; Jong, 2003; Abell, 2008; Hill, 2008;
Watson, Callingham & Donne, 2008; and Toerien, 2011) have made serious attempts to
develop models to measure teachers’ PCK in mathematics and the sciences. These
researchers
have largely been challenged by the difficulties the models present in
distinguishing the boundaries that make up the various constructs (Graham, 2011). These
difficulties include the changeable nature of PCK, which makes it difficult to pinpoint
specific constructs of this category of teacher knowledge (Miller, 2006). In addition, because
of the numerous categories of knowledge that could be integrated into PCK, differences may
exist in the boundaries of a PCK construct (Hill et al., 2008); and indeed because teachers,
like learners, construct their own knowledge, there is every likelihood that there will be
individual examples of teacher PCK. It is precisely because of these constraints that research
on PCK development has not always been as straightforward as researchers might have
hoped. A review of the literature indicates that the use of PCK in research and for methods
of data collection and analysis has mostly taken two forms (Shulman, 1986; Van Driel, 1998;
Penso, 2002; Jong, 2003; Cazorla, 2006; Abell, 2008; Hill, 2008; Watson, Callingham &
Donne, 2008; and Toerien, 2011). The first form has to do with research on PCK as a
category of teachers’ knowledge, that is, knowledge specifically constructed by teachers and
yet distinctly different for each subject matter content area. The second form involves
research using PCK as a theoretical framework, which is based on a number of assumptions,
as we shall see later. The fundamental difference between these two forms of using PCK in
research is that while the first entails trying to identify or measure PCK, the second utilises
the assumption that PCK exists, in order to examine other aspects of teacher knowledge
(Miller, 2006). In this study, the interest was in first determining teacher PCK in the context
of teaching school statistics, which is assumed to exist, and second in determining the way in
which it (PCK) is developed and used in teaching school statistics topics. To this end, the
study used PCK as a theoretical framework, consisting of teacher subject matter content
knowledge, pedagogical knowledge, and knowledge of learners' conceptions and learning
difficulties to explore the main research questions based on a number of assumptions.
8
The initial model of PCK, which was supported by several studies (eg Shulman 1987), tagged
PCK as the specific teacher knowledge that allowed a teacher to more thoroughly understand
how to transform content knowledge into a more conceptually accessible version for students
or learners. As explained by Shulman (1987) PCK results from the blending of content
knowledge and pedagogical methods. Thus, it is a widely accepted belief that PCK represents
the category of knowledge that is needed for a novice teacher to mature into an expert
(Bodner & Orgill, 2007). Shulman’s (1987) vision and Ball et al.’s (2008) description of
teacher knowledge as an amalgam of categories of knowledge, including content, curricular,
pedagogical, and student knowledge, and PCK,
has virtually compelled many teacher
education programmes to create new instructional
activities for improving classroom
practice. This same vision of enriching classroom practice has provided a focus on education
research. Unfortunately, PCK, because of its nebulous nature, remains a category of
knowledge that is difficult to isolate and research (Miller, 2006). Nevertheless, it provides a
starting point for researchers who wish to collect and analyse data on other aspects of teacher
knowledge. In this study the teachers’ classroom practice in statistics was therefore
investigated in a series of lesson observations, in order to explore what PCK exists and how
the participating teachers demonstrated their PCK in the context of teaching statistics in
school mathematics. The first consideration was that identifying the category(ies) of
knowledge that the teacher has, as defined, in the teaching of statistics would yield
information about teacher’s PCK and how it is developed and used during classroom practice.
It was mentioned earlier that the use of PCK as a theoretical framework has provided
researchers with a new perspective for collecting and analysing data about teacher knowledge
or cognition (Jong, 2003; Rollnick et al., 2008; Toerien, 2011). The use of PCK as a
theoretical framework allows researchers to focus on specific questions about a teacher’s
knowledge base and is founded on a series of assumptions. Miller (2006) has indicated that
PCK embodies an epistemological approach to understanding teacher knowledge. Precisely
for this reason, in this study, the teachers’ PCK in statistics teaching, and the way in which
they developed it, was conceptualised as comprising content knowledge, pedagogical
knowledge, and knowledge of learners' preconceptions and learning difficulties in the context
of teaching school statistics. These central categories of teacher knowledge were used as the
theoretical framework that provided a guide for data collection, analysis and discussion of
what and how PCK in statistics teaching was developed.
9
Assumptions of the study
Based on the above considerations, in this study the use of PCK as a theoretical framework
was built on the following assumptions, as summarised (Miller 2006).
¾ PCK represents a category of teacher knowledge that is the essence of an
expert teacher in a specific topic (Miller, 2006), in this case in school statistics
teaching. In this study, the blending of subject matter content knowledge,
pedagogical knowledge, and knowledge of learners' preconceptions and
learning difficulties was used to describe the PCK of the participating teachers.
¾ PCK provides a framework that can be used to describe the origin of this
critical teachers’ knowledge (Miller, 2006). In other words, PCK represents an
epistemological approach, to constructing teaching knowledge.
¾ PCK is a constructivist process and therefore a continually changing body of
knowledge. Teachers, like learners, construct their own knowledge and in this
study it is assumed that the development of PCK is a continuously modifying
unit, beginning with teacher preparation programmes, evolving through
teaching experience and assimilating and accommodating professional
development opportunities.
¾ Identifying and measuring PCK constructs can be achieved by using
instruments designed for that purpose. In this study, the components of
PCK were assessed using multiple assessment strategies, which
include concept mapping, teacher interviews, teacher questionnaires,
lesson observation, written classroom activity reports and document
analysis. According to Shulman (1986), PCK is a specific category of knowledge that goes beyond the
knowledge of subject matter per se to include the dimension of subject matter knowledge for
teaching. It refers to teachers’ interpretations of subject matter in the context of facilitating
learning. In consequence, it has been argued that PCK is one of the seven categories in
Shulman’s (1986) categorisation of a knowledge base for teaching. The key elements of
10
Shulman’s conception of PCK are:
i)
Knowledge of the representation of the subject matter for teaching
ii)
Knowledge of relevant instructional strategies
iii)
Knowledge of learners’ conceptions (preconceptions and misconceptions)
iv)
Knowledge of learners’ learning difficulties
For the purpose of this study, these four elements appear to be most appropriate in defining
the PCK that may be used for teaching statistics in school mathematics, namely subject
matter content knowledge; knowledge of teaching (pedagogical knowledge); knowledge of
learners’ conceptions (preconceptions and misconceptions); and knowledge of learners’
learning difficulties. These four elements cover the views and constructs of PCK used by
various researchers in this domain, such as Jong (2003), Shulman (1986), Jong et al. (2005),
Halim and Meerah (2002), Rollnick et al. (2008), Hill (2008) and Toerien (2011).
For the construct of PCK, the working definition is that PCK is an amalgam of subject matter
content knowledge, pedagogical knowledge (instructional skills and strategies), knowledge of
learners’ conceptions and knowledge of learners’ learning difficulties. In this study, the
researcher’s intention was to determine the PCK that competent teachers use in teaching
statistics by observing the PCK that such teachers demonstrate in the classroom. It is assumed
that because such teachers are considered competent and have experience in teaching
mathematics, they will be able or will be likely to integrate content knowledge and
pedagogical knowledge in ways that contribute to the development of the PCK used for
teaching statistics (Jong, 2003). To this end, the development of PCK was inferred from the
teacher interviews, questionnaires, written reports, document analysis and lesson observation.
1.7.1
Subject matter content knowledge
According to Manouchehri (1976), subject matter content knowledge consists of an
explanatory framework and the rules of evidence within a discipline. The subject matter
content knowledge of prospective mathematics teachers is acquired primarily during
disciplinary education (Jong, 2003). This knowledge consists of substantive content
knowledge and syntactic content knowledge (Barnes, 2007). Substantive content knowledge
refers ‘to the concepts, principles, laws, and models in a particular content area of a
11
discipline’. Syntactic content knowledge, by contrast, is the ‘set of ways in which truth or
falsehood, validity or invalidity are established’ (Schwab, 1978, cited in Shulman, 1986). In
practice, teachers should be able not only to define the acceptable truths for learners in a
domain, but to explain, in theory and in practice, why these truths are worth knowing and
how they relate to other propositions, within the discipline and outside it.
Both types of subject matter knowledge (substantive and syntactic) are needed for teachers’
development of PCK, because they help to create an adequate understanding of the nature of
the subject matter and beliefs about how it should be taught (Jong, 2003). It is therefore
assumed that mathematics teachers with good PCK have both types of subject matter content
knowledge and are able to apply this knowledge in making the topic understandable to
learners. This assumption is given empirical support by Wu (2005), who indicated that
teachers with good PCK have a firm command of subject matter knowledge and are able to
design mathematics instructional material that allows learners to grasp what they teach. Muijs
and Reynolds (2000) referred to these teachers as effective teachers.
Other scholars, such as Carpenter, Fennema, Petterson and Carey (1988), Even (1993),
Manouchehri (1997), Van Driel et al. (1998), Halim and Meerah (2002), Tsangaridou (2002),
Viri (2003) and Hill (2008), have studied the influence of subject matter knowledge on the
PCK of pre-service, novice and expert teachers. These studies revealed that teachers’ content
knowledge goes a long way towards determining the level of teachers’ PCK. The subject
matter content knowledge is one of the components of PCK that will be assessed in this
study.
1.7.2
Pedagogical knowledge
Cochram et al. (1993) define pedagogical knowledge as knowledge about teaching. VistroYu (2003) defines it as the knowledge used for teaching, particularly expertise in teaching
techniques, psychological principles, classroom management, and teaching and learning
processes. Following these definitions, pedagogical knowledge is believed to be the kind of
information that a teacher needs and uses to perform everyday teaching tasks, involving
teaching styles and strategies, classroom management and teaching and learning processes
relating to learners in the classroom. Research findings by Rollnick et al. (2008), Jong et al.
(2005) and Vistro-Yu (2003) show that a mathematics teacher with adequate pedagogical
knowledge is able to design good teaching and learning strategies and manage the classroom
12
and other instruction and learning processes. This constituent framework seems appropriate
for defining the construct of pedagogical knowledge, as it describes in operational terms what
the teacher needs to do to create an environment that is conducive to learning. In this study,
the focus was on the instructional skills and strategies used for teaching statistics in school
mathematics.
To this end, the pedagogical knowledge of mathematics teachers was assessed by examining
their lesson planning and implementation, questionnaires, written reports, and interviews, in
order to probe the way in which competent teachers develop their pedagogical knowledge
and use it in the instruction and learning process.
1.7.3
Knowledge of learners’ conceptions in statistics teaching in school
mathematics
Learners’ conceptions in statistics in school mathematics consist of preconceptions and
misconceptions. A mathematical misconception is a belief or idea that is based on incorrect
or erroneous information about a given mathematical concept (Olivier, 1989). According to
Olivier (1989), most mathematical misconceptions arise because of pre-existing concepts or
preconceptions in the mind of the learner or the teacher. Misconceptions can occur when an
attempt is made to link preconceptions and new knowledge to be learned. Olivier (1989)
argues that misconceptions play a key role in understanding a new concept. The role of the
mathematics teacher in resolving mathematical misconceptions is usually to develop some
form of teaching and learning approach, such as teacher-learner or learner-learner discussion,
communication, reflection, and negotiation of meaning, that addresses the missing concept
(Penso, 2002; Cazorla, 2006). Through these approaches, the mathematics teacher may be
able to get to the root of the misconception.
Cazorla (2006) for example reported that misconceptions and the way in which mathematics
lessons are taught are among the factors that cause learning difficulties. According to her,
most statistics teachers do not have adequate knowledge of the school curriculum and the
approaches needed to teach and learn statistics, which can result in poor content delivery in
the classroom situation. Jong (2003) noted that in order to identify and resolve
misconceptions and learners’ learning difficulties during classroom practice, the teacher
could use convergent and inferential techniques. Convergent and inferential techniques are
data-collection systems that entail developing questions for a topic in short-answer and
13
multiple-choice formats to probe the preconceptions and misconceptions of learners (Jong,
2003). The teachers’ written reports and the learners’ notebooks may help to identify where
the learners’ learning difficulties lie (Jong, 2003; Jong et al., 2005, Penso, 2002; Van Driel et
al., 1998).
The participating teachers in this study will be examined to determine whether they have
prior knowledge of statistics as they teach the assigned topic through lesson planning and
implementation.
1.7.4
Knowledge of learners’ learning difficulties
Penso (2002) reports that learning difficulties may stem from the way lessons are taught. For
example, learning difficulties may arise from the content of the lesson, lesson preparation and
implementation and the learning atmosphere (Penso, 2002). Other factors include
misconceptions that learners and teachers have about a topic, as well as cognitive and
affective characteristics of learners. According to Penso (2002), ‘learners consider their
learning difficulties to be the result of conditions that existed prior to the process of teaching,
as well as those existing in the course of teaching’.
In this study, the ways in which the teachers identified and addressed the learning difficulties
that learners encountered during classroom practice were determined in lesson observation.
From the above discussion, subject matter content knowledge, pedagogical knowledge
(instructional
skills
and
strategies),
learners’
conceptions
(preconceptions
and
misconceptions) and learners’ learning difficulties were used to conceptualise the construct of
PCK for teaching school statistics. These frameworks were derived from the model proposed
by researchers such as Shulman (1986), Van Driel (1998), Jong (2003), Cazorla (2006),
Penso (2002), Abell (2008), Hill (2008) and Toerien (2011), as discussed in sections
1.7.1 1.7.4. The selection of these components of PCK was based on the assumption that
PCK is dynamic, topic specific, and transformative, and can be measured using these
frameworks (Corrigan, 2008). While the subject matter content of the participating teachers
was measured using a conceptual knowledge exercise, concept mapping, interviews and
lesson observation, instructional skills and strategies were assessed using lesson observation,
questionnaires, interviews, written reports and document analysis. Lesson observation,
questionnaires, written reports and reviews of teachers and learners’ portfolios, as well as
14
lesson plans and learners’ workbooks, were used to assess the teachers’ knowledge of
learners’ preconceptions and learning difficulties in statistics teaching. The roles of each
instrument in measuring the individual component are described in Section 3.5.1.
To summarise, subject matter content knowledge, pedagogical knowledge, and learners’
conceptions and learning difficulties were used to conceptualise the PCK needed for teaching
statistics in school mathematics.
1.8
Scope of the study
This study explored how selected mathematics teachers at high-school level develop PCK in
statistics teaching. The focus was on teachers who were teaching mathematics in accordance
with the NCS Curriculum (now called Curriculum and Assessment policy Statement (CAPS))
for Mathematics at high schools in Tshwane North Education District in South Africa. These
teachers were selected as participants for this study based on the performance of their learners
in the public Senior Certificate Examination and on being recommended as competent
teachers by principals, peers, and mathematics specialists at the Department of Basic
Education (DoBE). Since PCK is topic specific (Corrigan, 2008), data were collected during
statistics lessons by means of lesson observation. The participants in this study were few,
because of the criterion used, namely a pass rate of 70%, and because participation was
voluntary.
1.9
Criteria for selecting the topic
The concept of statistics is defined by Otumudia (2006) as the science of collecting,
organising, and analysing data for any given purpose. Statistics helps us to reduce large and
scattered data to an understandable level, thereby enabling us to make decisions in the face of
uncertainty (Otumudia, 2006).
Statistics is taught as part of the mathematics curriculum under the rubric of ‘data handling’.
Data handling is one of the four major topics in the Curriculum and Assessment Policy
Statements (CAPS) (DoBE, 2011). The reasons for including data handling in the new
curriculum are:
i)
Basic statistical knowledge is necessary for all kinds of data interpretation, as people
15
encounter a great deal of categorical and numerical observations that should be used to
guide decisions (DoE, 2006).
ii)
Data handling helps one to build critical thinking, to understand reality, and to be able
to participate in social actions.
iii)
Statistics and probability are useful in daily life and play an instrumental role in other
disciplines, such as economics, engineering, and medicine (Franklin & Mewborn,
2006).
iv)
There is a need for basic stochastic knowledge in many professions, and statistics plays
an important role in developing critical thinking that help in the development of this
type of knowledge (Innabi, 2002).
For these reasons, the learning outcomes require learners studying statistics to be able to
collect, organise, analyse, and interpret data to establish statistical and probability models for
solving related problems (DoBE, 2011).
1.10
Definition of terms
In this section, some of the terms that are used to describe how mathematics teachers develop
PCK for statistics teaching are defined operationally.
•
National Curriculum Statements (NCS)
The National Curriculum Statements (NCS) are guidelines that state what each learner should
achieve in terms of learning outcomes and assessment standards by the end of each grade. In
this study, the NCS for Mathematics is used to describe the curriculum for mathematics as the
subject that is taught in Grades 10 to 12.
•
Curriculum and Assessment Policy Statement (CAPS)
The National Curriculum and Assessment Policy Statement is a ‘single, comprehensive, and
concise policy document, which replaced the Subject and Learning Area Statements,
Learning Programme Guidelines and Subject Assessment Guidelines for all the subjects
listed in the National Curriculum Statement Grades R – 12’ (DoBE, 2012).
•
Pedagogical content knowledge (PCK)
16
The construct of PCK constitutes an amalgam of subject matter content knowledge,
pedagogical knowledge (instructional skills and strategies), knowledge of learners’
conceptions and knowledge of learners’ learning difficulties. In this study, PCK is used to
describe and measure the way mathematics teachers combine subject matter content
knowledge and pedagogical knowledge, as well as use their knowledge of learners’
preconceptions and learning difficulties to carry out effective teaching during classroom
practice.
•
Pedagogical knowledge
Pedagogical knowledge is that knowledge that a teacher needs and uses to perform everyday
teaching tasks, involving instructional skills and strategies, and classroom management and
teaching and learning processes relating to learners in the classroom (Vistro-Yu, 2003).
Pedagogical knowledge is used to define the construct of PCK in statistics teaching in this
study. Specifically, the instructional skills and strategies will be used to describe the
pedagogical knowledge in statistics teaching in this study.
•
Conceptions in the teaching and learning of statistics
Conceptions in teaching and learning statistics consist of preconceptions and misconceptions.
A preconception is regarded as the prior knowledge of a given topic with which learners
come to the class (Olivier, 1989) and is used as such in this study. It is manifested during
lesson observation. A misconception can occur as a result of a pre-existing concept. Both
preconceptions and misconceptions can contribute to learners’ learning difficulties in
classroom practice. The term ‘misconception’ was used to describe the learners’ beliefs or
notions that were based on incorrect or erroneous information about a given statistical
concept demonstrated during classroom practice. Teachers’ knowledge of learners’
conceptions in learning statistics was used to describe the PCK that was likely to be used for
teaching statistics in school mathematics.
•
Competent mathematics teachers
In this study, competent mathematics teachers were identified based on their learners’ final
results in mathematics in the public senior certificate exam and recommendations made by
17
principals, peer teachers and subject experts at the Department of Education. Although being
competent may not necessarily mean that the teachers are knowledgeable or expert in
statistics, they were able to help their learners to do well in their final Senior Certificate
Examination in Mathematics. The teachers were observed while teaching school statistics in
order to determine how they develop their PCK.
•
Conceptual knowledge
Conceptual knowledge involves an understanding of mathematical ideas and concepts,
as well as the interrelationships among these concepts. It consists of the ability to
identify and apply principles, facts and definitions, and to compare and contrast
related concepts (Engelbrecht & Potgieter, 2005). In this study the conceptual
knowledge approach involves the use by the teacher of mathematical ideas, principles,
facts and definitions to explain mathematical concepts and their relationships during
the teaching and learning of a particular topic.
•
Procedural knowledge
Procedural knowledge is a formal symbolic representation system of a given
mathematical task using algorithms, or rules, to complete the mathematical tasks
(Star, 2002). In practice, it means for the teacher the use of particular rules, algorithms
or procedures to complete a given task without necessarily providing an explanation
underpinning the rules or procedures used. For example, the construction of statistical
such as bar graph, histogram, ogive and scatter diagrams requires that one should first
draw the axes, choose the scale, label the axes, plot the points and join the line of best
fit (Leinhardt et al, 1990). The four participating teachers followed this procedure
during their lessons on bar graphs, histograms, ogives and scatter diagrams. This
teaching approach essentially uses what is referred to in this study as a procedural
knowledge approach.
•
Document analysis
Document analysis is a technique used in this study to gather information. It describes the act
of reviewing the documentation of comparable school systems in order to extract pieces of
information that are relevant to the current research project. Hence it is sometimes regarded
as a research project requirement. In this study, document analysis was used to extract
information about teaching and learning of statistics from the NCS for Mathematics, teacher
and learner portfolios, and learners’ class workbooks.
18
1.11
The chapter structure of the thesis
The study is divided into six chapters. Chapter 1 presents the introduction and background of
the study and the way in which the background relates to the problem under investigation.
The context in which the study took place is described.
PCK, as one of the forms of knowledge needed to implement the curriculum, was defined
from four perspectives, namely content-specific knowledge; content-specific instructional
strategies; knowledge of learners’ conceptions of statistics teaching and learning; and
knowledge of learners’ learning difficulties. The chapter then presents the guiding research
questions and theoretical framework based on the purpose of the study and the statement of
the problem. The key concepts in this study are highlighted and discussed. The chapter
concludes with a brief discussion of the structure of the thesis.
Chapter 2 focuses on the literature review, which captures the empirical and theoretical
aspects related to the process of PCK development and how it is used in classroom practice to
teach mathematics and science. The literature review derives its focus from the National
Curriculum Statement for Mathematics, theoretical framework and the research questions,
which seek to describe the way in which competent mathematics teachers develop PCK in
statistics teaching. Chapter 2 is divided into two sections. The first section discusses literature
about the content of statistics according to NCS and research on the teaching of statistics in
school mathematics. The second section discusses the models of capturing PCK,
conceptualisation of PCK and techniques for measuring PCK.
Chapter 3 discusses the methodology of the study. It is argued that a rich description of data
comes from using several strategies of investigation, data collection, and data analysis. The
chapter describes the methodological plans for the study, the pilot study, the participants, the
research activities, and the various instruments used in the collection of data. The validity and
reliability of the instruments are also discussed in this chapter.
Chapter 4 presents the results of the data collection discussed in chapter 3. The first
presentation concerns the quantitatively analysed data, and the second concerns the
qualitatively analysed data. The latter relies on the quantitative data that has been analysed.
While the quantitative data are derived from the conceptual knowledge exercises, the concept
mapping exercises, and the results of the schools from which the participants were selected,
19
the qualitative data are derived from the interviews, lesson observations, free-response
questionnaires, teachers’ written reports and documents related to teachers’ guides, and
learners’ portfolios, mathematics workbooks, and textbooks. In this chapter, the guiding
research questions and the theoretical framework are revisited in order to determine how
competent mathematics teachers develop their PCK in statistics teaching.
Chapter 5 contains a discussion of the results, based on the results of the previous chapter.
The guiding research questions are revisited. In line with the theoretical framework, the
chapter presents a discussion focusing on the teachers’ PCK profiles and how the data help to
answer the research questions in order to determine how the mathematics teachers developed
their pedagogical content knowledge in statistics teaching.
Chapter 6 presents a summary, the conclusions of the study, and recommendations and
suggestions for further research.
1.12
Summary of chapter
This chapter provided insight into the research orientation used in this study, in an attempt to
make the reader conversant with the research project. The chapter began with an introduction
to the NCS and the learning outcomes for statistics in school mathematics. The knowledge
that the teacher needs to implement the curriculum effectively was highlighted from three
perspectives, namely subject matter content knowledge, pedagogical knowledge, and PCK.
The introduction was followed by an elucidation of the problem of the study, a statement of
the research problem, the aims of the research, the research questions, the significance of the
study, the scope of the study, and the theoretical framework that guided the study. The key
concepts used in this study were then defined and discussed, and the chapter concluded with a
discussion of the criteria for selecting the topic, as well as the chapter structure of the thesis.
20
CHAPTER 2
2.0
LITERATURE REVIEW
2.1
Introduction
This chapter focuses on the literature review, which tries to address the empirical and
theoretical issues related to teachers’ PCK development and its use in mathematics and
statistics teaching. The discussion about the process of PCK is derived from a review of the
NCS for mathematics and statistics teaching and the research questions guiding the study.
Studies on teaching statistics in school mathematics and the models for capturing PCK are
discussed. The techniques of studying PCK are highlighted and studied in order to justify the
validity of the instruments used to investigate PCK. The chapter concludes with a summary
of the theoretical framework that allows for the development of the research instruments, data
analysis and results.
2.2
National Curriculum Statements for Mathematics and Statistics
The National Curriculum Statement (NCS) for Mathematics is based on the nature of the
discipline and societal expectations of learners of mathematics (DoE, 2009). Mathematics is a
subject that enables creative and logical reasoning about problems in the physical and social
world, and in the context of mathematics itself (DoE, 2009:9). From this, mathematics is seen
as a human activity that deals with patterns, problem solving, and logical thinking, in an
attempt to understand the world and to make use of that understanding (Lebeta, 2006).
According to the views of the Department of Education (2009) and Lebeta (2006), it may be
concluded that ‘mathematics is part of day-to-day human experiences and relates to human
activities that use features of one natural object as a tool for acting on other objects. This
means that mathematics is an organic activity’. According to Davydov (1999), human activity
is linked to conceptual activity. The purpose of mathematics is to demonstrate how human
activity is linked to conceptual activity. Therefore, ‘knowledge in mathematical science is
constructed by establishing descriptive, numerical and symbolic relationships that are based
on observing patterns, using rigorous logical thinking that can lead to theories of abstract
relations’ (DoE, 2009). By implication, mathematical knowledge can help learners to engage
in problem solving to understand the world, and they can use that understanding in their daily
lives. Hence, the subject statement for mathematics for Grades 10 to 12 expects learners to
21
expand on their understanding of Learning Outcome 4 (LO4) of the NCS under the category
‘Data handling and probability’ (DoE, 2007:22), through appropriate teaching and learning of
the topic in the classroom context. This learning outcome ‘requires learners to be able to
collect, organise, analyse, and interpret data, in order to establish statistical and probability
models to solve related problems with a focus on human rights issues, inclusivity, and current
matters involving environmental and health issues’ (DoE, 2009:10). What, then, is the
purpose of mathematics, according to the NCS?
According to the NCS (2009:11), the purpose of mathematics is to provide powerful tools:
•
To analyse situations and arguments, make and justify critical decisions, and take
transformative action, thereby empowering people to work towards the reconstruction
and development of society
•
To develop equal opportunities and choices
•
To contribute towards the widest development of society’s cultures, in a rapidly
changing, technological, global context
•
To derive pleasure and satisfaction through the pursuit of rigour, elegance, and the
analysis of patterns and relationships
•
To engage with political, organisational and socio-economic relations (DoE, 2009:11)
However, the focus of this study is on statistics, which is part of the mathematics curriculum.
Research reports by Gattuso (2006) show that there is a link or relationship between
mathematics and statistics. For example, linear function is used in describing the relationship
between two variables in scatter plots. Using the stem-and-leaf diagram, one can distinguish
between units and tens in mathematics. And in the workplace, statistics is used in
representing the records of employees’ weekly, monthly and yearly attendance at work on a
frequency table and statistical graphs. That is why it is important that mathematics teachers
understand this relationship, so that it can be addressed in the teaching and learning situation
(DoE, 2009).
For many teachers, the relationships are not clear. They face difficulties in teaching statistics
and addressing the relationships between mathematics and statistics in classroom practice
(DoE, 2010). As early as 1988, Garfield and Ahlgren (1988) reported that although statistics
22
is related to the learning of mathematics and other disciplines, a large proportion of learners
do not understand many of the basic statistical concepts they have studied. The authors
reported that ‘inadequacies in prerequisite mathematics skills and abstract reasoning’ are part
of the difficulties encountered by the learners of statistics. Poor learner performance in
statistics was also noted at the joint conference of the International Commission for
Mathematics Instruction and the International Association for Statistics Educators
(ICMI/IASE, 2007).
2.3
Research on teaching statistics in school mathematics
The important role of statistics in mathematics education and other disciplines has now been
recognised worldwide. This was confirmed by the introduction of statistics in school
mathematics in the school curricula at all levels in South Africa and elsewhere (DoE, 2009).
However, recent research on teaching of statistics in school mathematics shows that learners
encounter difficulties in learning the subject (Godino et al., 2011).
Baker, Corbett and Koedinger (2001) observed that learners are often confused about the
construction of bar graph and histogram. According to these authors, most learners construct
a histogram in the same way as a bar graph. The authors noted that in the stage of learning
how to construct a histogram, learners transferred their existing knowledge about a bar graph
to the construction of a histogram, instead of using knowledge specific to the target
representation. And because learners were already familiar with bar graph construction, they
found it easy to construct a bar graph instead of histogram (Baker et al, 2001).
Meletiou-Mavrotheris and Lee (2002) note that learners perceive histograms as twodimensional graphs that must have two variables and thus tend to interpret a histogram as
two-variable scatter plots. In addition, learners tend to perceive histograms as displays of raw
data on the Y-axis with each bar standing for individual observation and with individual cases
on the X-axis. These authors reported that when comparing two histograms with regard to
their variability, learners used the vertical axes of the histogram instead of the horizontal axes
to compare their variability or spread (Meletiou-Mavrotheris & Lee, 2002).
Baker et al. (2001) extended this research to include the construction and interpretation of
statistical graphs with emphasis on scatter plots and stem-and-leaf. Their reports show that
23
the axes of a scatter plots were drawn by the learners as if a bar graph was to be represented
and plotted the points on the wrong axes. Consequently, a misinterpretation was obtained
from a wrongly constructed scatter plot.
Other research studies (NCTM, 2007; Baker et al., 2001, Cazorla, 2006 and DoBE, 2012)
attributed learners’ learning difficulties to the way teachers taught the construction and
interpretation of stem-and-leaf diagrams. The authors noted that although learners can read
and represent stem-and-leaf diagrams, they were unable to interpret them because they had
not been exposed to the types (varieties of ways) of stem-and-leaf representation.
Nicholson and Darnton (2005) researched the challenges for the classroom teacher in
teaching statistics. In an analysis of questions used in statutory national tests, learners’ scripts
were used to collect data on their reasoning processes and learning difficulties. The results of
an analysis of the questions and scripts at the early stage in the primary school were
compared with the difficulties seen at the later stage of secondary statistics. The findings of
this study show that pupils at the early stage struggle to articulate their reasoning processes
explicitly. Furthermore, teaching and learning at the later stage of their secondary
examination were based on computational accuracy and procedural competence in statistics,
and less time was spent on interpretational skills. The implication of these findings is that
mathematics teachers who are not familiar with the common difficulties and misconceptions
may not be able to help learners to overcome their learning difficulties in statistics and
achieve a deeper understanding of core concepts (Nicholson & Darnton, 2005).
Mavrotheris and Stylianou (2003) observed that one of the sources of learning difficulties in a
statistics classroom is that most mathematics teachers are too formalistic in their approach to
the subject. The authors noted that statistics lessons are presented in rigidly established
bodies of mathematical knowledge without any reference to the real-world context
(Mavrotheris & Stylianou, 2003). Formalist ways of teaching have led to educators failing to
convey to the learners the relationship between knowledge they acquire in the statistics
classroom and its uses in everyday life (Mavrotheris & Stylianou, 2003). For example,
learners were taught first to build a cumulative frequency table, and construct an ogive by
drawing the axes, labelling the axes, plotting the points and joining the line of best fit. During
interpretation and analysis, values were read off from the vertical and horizontal axes without
24
being linked to the learners’ real world (Libman, 2010). Hence, learners had difficulties in
understanding what the teacher had taught using the formalistic approach.
Watson, Callingham and Donne (2008) carried out research on establishing PCK for teaching
statistics from Grades 1 to 12. The PCK of 42 teachers selected as part of a professional
learning programme in statistics was examined. The results of the Rasch analysis to obtain a
measure of teacher ability levels in relation to PCK indicate that teachers who did not
respond appropriately to the survey items often missed or left out those items that required a
response to a specific student misunderstanding (Watson, Callingham & Donne 2008). The
inability of the teachers to respond to specific student misunderstanding could mean either
that they were not able to move students towards a higher level of statistics understanding or
to design instructional interventions to address students’ learning difficulties. This study
represents an initial attempt to establish the nature of teachers’ demonstrated PCK in teaching
school statistics.
The intention of the researcher through this study is to determine whether the participating
teachers are aware of their learners’ difficulties with statistical graphs and the means used by
them to elicit these difficulties. PCK is seen as a relevant construct for this study as teachers’
topic-specific content knowledge influences what is taught in the classroom context. It
therefore becomes necessary to explore the PCK of a mathematics teacher who demonstrates
good content-specific knowledge (Godino, Batanero & Font, 2011) to see how this teacher’s
PCK is enacted while teaching these difficult topics.
2.4
Assessing teachers’ PCK
2.4.1
Description of PCK
The concept of pedagogical content knowledge (PCK) was introduced by Shulman (1986) in
a paper in which he argued that research on teaching and teacher education ignored questions
dealing with the contents of lessons, the questions asked, and the explanations offered. As
indicated in the theoretical framework of this study, PCK goes beyond knowledge of the
subject per se to encompass the dimension of subject matter knowledge for teaching. It refers
to how the teacher interprets the subject matter knowledge in the context of facilitating
learning.
25
Shulman (1986), while categorising a knowledge base for teaching, noted that the way in
which the subject matter is presented and formulated is a key element in the conceptualisation
of PCK. According to him, this knowledge could originate from research or teaching practice.
Other elements in Shulman’s categorisation of the knowledge base for teaching are awareness
of strategies that may be fruitful in reorganising the understanding of learners, and learners’
preconceptions and misconceptions about a particular topic.
In the two decades since Shulman introduced the concept of PCK there have been a number
of studies on the subject. Various scholars across the discipline have elaborated on Shulman’s
work and proposed different conceptualisations of PCK (Grossman, 1990; Marks, 1990;
Cochram et al., 1993; Van Driel et al., 1998; Magnusson, Krajcik & Borko, 1999; GessNewsome and Lederman, 2001; Barnett & Hodson, 2001; Jong, 2003; Halim & Meerah,
2002). This amplification is in terms of what they include or do not include in their
conceptualisations of PCK.
Grossman (1988) developed and expanded the definition of PCK. Her definition is based on
four central components: knowledge of learners’ understanding; the curriculum; instructional
strategies; and the purpose of teaching. Knowledge of learners’ understanding refers to how
the learners comprehend what is taught. In other words, how do learners understand the
subject matter being presented to them? The curriculum pertains to the content of the subject
matter, as contained in it. Knowledge of instructional strategies constitutes understanding of
the stratagems employed in teaching the subject. The purpose of teaching is to achieve the
learning outcomes, as outlined in the curriculum. Using these components, Grossman (1988)
examined the influence of teacher education on knowledge growth. The findings regarding
the impact of teacher education on knowledge growth demonstrate that teacher education can
influence knowledge growth by teachers.
Teacher education involves the disciplinary tutoring through which the subject matter
knowledge and pedagogical knowledge can be acquired. This education can provide an
opportunity to acquire more knowledge and growth if the teacher continues to practise in the
particular discipline (Grossman, 1988). The influence of teacher education on knowledge
growth is related to this study in the sense that one can speculate that the disciplinary
education acquired by teachers could influence the way in which their PCK is developed and
used for teaching statistics in school mathematics, hence the need to examine and assess the
26
level of teachers’ subject matter content knowledge, as already indicated. However, in the
context of delivering a particular curriculum (DoE, 2007), the model fails to indicate any
specific programme and how it influences the teachers’ knowledge and its uses during
classroom practice (Ibeawuchi, 2010).
Based on an explicit constructivist view of teaching, Cochram et al. (1993), in their research
on PCK as an integrative model for teacher preparation, renamed PCK ‘pedagogical content
knowing’ (PCKg), to acknowledge the dynamic nature of knowledge development. In their
model, PCKg is conceptualised far more broadly than in Shulman’s view. They define PCKg
as ‘a teacher’s integrated understanding of four components of pedagogy, subject matter
content knowledge, learner characteristics and the environmental context of teaching’
(Cochram et al., 1993). According to these authors, PCKg is generated as a synthesis of the
simultaneous development of these four aspects in the context of the integrative model of
teaching. Following this argument, it means that the components of PCK, as highlighted
above, do not exist independently of one another. In this study, however, the components of
PCK were captured individually during classroom practice. Even though the elements of
PCK do not exist independently of one another as conceptualised, it is still seen as an
amalgam of these components during classroom practice. PCK is individualistic, tacit, and
ever changing with time and experience (Miller, 2007).
But according to Lee and Luft (2008), there are two models of PCK, integrative and
transformative. In the integrative model, the PCK components exist separately, and at the
beginning of teachers’ careers they enable teachers to rely on only one of the PCK
components to cope with teaching (subject matter content) (Lee & Luft, 2008).
Transformative PCK is held by experienced teachers who combine all the components of
PCK and convert it into classroom practice. Lee and Luft (2008) claimed that during teaching
it is difficult to distinguish subject matter knowledge or general pedagogical knowledge from
PCK, which means the components do not exist independently of one another. In this study,
based on notion of amalgam, the components of PCK can exist independent of one another or
together. The ways the teachers used them were established by attempting to describe the
PCK profiles of the participating teachers as evidence in their practice.
Van Driel, Verloop and De Vos (1998) conducted research on developing science teachers’
PCK, using classroom observation and interviews. According to them, the idea of integration
27
of knowledge components is also central to the way PCK is conceptualised by FernandezBalboa and Steel (1995). These authors identify five knowledge components of PCK: subject
matter, the learners, instructional strategies, the teaching context, and the teaching purpose.
Magnusson et al. (1999) presented a model of the relationship between the constituent
domains of PCK. According to them, subject matter knowledge (e.g. substantive knowledge,
and syntactic knowledge), pedagogical knowledge, knowledge of educational aims,
knowledge of the classroom, and context knowledge (e.g. knowledge of specific learners and
school characteristics) could be used to interpret PCK. In the teaching process, these domains
could be combined (Rollnick et al., 2008) to provide effective teaching and promote learners’
understanding of the lesson.
Barnett and Hodson (2001), in their research on how to understand what science teachers
know, considered PCK a constituent of pedagogical context knowledge, together with other
components. These other components were academic knowledge, classroom knowledge, and
professional knowledge. But the components of PCK are not always clear and consistent;
rather they look blurry; and the development of a teacher’s PCK is not linear, but advances
from different angles (Loughran et al., 2004).
Although different researchers have varying opinions about the conceptualisation of PCK,
Jong (2003) and Van Driel et al. (1998) stated that these elements seem to be germane to any
conceptualisation of PCK with respect to a chosen content area
•
Knowledge of learners’ learning difficulties, conceptions, and misconceptions
concerning the topic
•
Knowledge of how to represent specific topics
Several scholars have researched PCK development, and their studies are concerned with
how a teacher uses his/her knowledge of the content that the learners are expected to learn
and the best approaches to employ to access that content; hence it is called the knowledge
base for teaching. A teacher’s PCK is therefore unique (Bucat, 2004) as it depends on how he
or she interprets learners’ preconceptions and learning difficulties and what the learners need
in order to understand the content being taught (Mitchell & Mueller, 2006). The development
28
of PCK is mutual and hence the development of one component influences the development
of another (Henze, Van Driel & Verloop, 2008). Hill et al. (2008) argued that the impact of
teachers’ PCK on learners’ learning was still to be proven, since there seemed to be a
relationship between the teacher’s PCK and what the teacher does in the classroom. So far,
these authors have agreed that the development of a teacher’s PCK is rooted in the classroom
and this could contribute to effective teaching and learning of statistics in school
mathematics.
The first component of PCK, namely knowledge of learners’ understanding and their
conceptions of a specific topic, helps teachers to interpret learners’ actions and ideas, as well
as plan effective instruction (Loughran, Mulhall & Berry, 2004; Halim & Meerah, 2002).
These authors argued that ignorance of learners’ misconceptions may be due to teachers’ lack
of content knowledge. The second component, knowledge of how to teach a particular topic,
refers to awareness of specific areas that are useful in helping learners understand specific
concepts. This involves knowledge of ways of representing specific concepts, in order to
facilitate learning (Halim & Meerah, 2002). This component of PCK, which aims to develop
learners’ conceptual understanding, seems necessarily dependent on having subject matter
knowledge relative to the concept being taught. Furthermore, ‘the PCK for representing
specific topics is a product of previous planning, teaching and reflecting’ (Halim & Meerah,
2002).
2.4.2
Teacher knowledge and PCK
According to Gess-Newsome (in Jong, 2003), all the various views of PCK can be
categorised as integrative or transformative. Where PCK is categorised as integrative,
knowledge of teaching is merely the integration of forms of teacher knowledge, such as
subject matter content knowledge, knowledge of learners’ learning difficulties, and
knowledge of learners’ preconceptions concerning a topic. In this integrative view, PCK is
seen as a mixture. In other words, ‘PCK does not really exist in its own domain, and teaching
is seen as an act of integrating knowledge of subjects, pedagogy and context’ (GessNewsome & Lederman, 2001). In classroom practice, knowledge of all these domains is
integrated by the teacher to create effective teaching and learning opportunities. Most teacher
education programmes that are organised in separate courses of subject matter, pedagogy, and
practice follow this model of teacher knowledge (Ibeawuchi, 2010).
29
In the transformative view (Jong, 2003), forms of teacher knowledge, such as subject matter
knowledge, pedagogical knowledge, and contextual knowledge are transformed into a new
form of knowledge such as understanding of a concept. In this view, PCK is seen as a
compound. This model supports teacher education programmes that contain integrated
courses and allow prospective teachers to quickly develop the required skills and knowledge.
The integrative view and the transformative view can be considered opposite ends of the PCK
spectrum (Jong, 2003). In this study, it is assumed that the transformative view was used by
the participating teachers during classroom practice for teaching statistical graphs because the
teachers uses the conceptual knowledge approach to describe the concept of histogram, ogive
and bar graph which the learners appear to have understood.
Recently the statistics education community’s attention has been drawn to the statistical
knowledge for teaching (SKT) measures by scholars such as Hill, Blunk, Charalambous,
Lewis, Phelps, Sleep and Ball (2008). According to these authors, statistical knowledge for
teaching included statistical information that is common to individuals working in diverse
professions and the subject matter knowledge that supports such teaching, for example why
and how a statistical procedure works, how best to define a statistical term for a particular
grade level, and the particular content (Hill et al., 2008). To these authors, the impact of
teachers’ PCK on learners’ learning had yet to be proven, but there seemed to be a
relationship between a teacher’s PCK and what the teacher did during classroom practice.
Following these arguments, the development of PCK is explored in the classroom, and this
can contribute to effective teaching and learning.
Toerien (2011) conducted preliminary research on the development of PCK of in-service
science teachers and conceptualised PCK as including subject matter content knowledge, the
context of the school, knowledge of the curriculum, and teachers’ pedagogical knowledge.
Using semi-structured interviews and lesson observation, Toerien (2011) noted that these four
components could be used to investigate the development of PCK of in-service science
teachers in the classroom context.
In looking at how various researchers have conceptualised PCK, it appears that investigating
PCK may not always be a straightforward matter, because of its unarticulated and tacit
nature. Jong et al. (2005) argued that investigating PCK development is a complex process,
30
because PCK is determined, among other things, by the nature of the topic, the context in
which the topic is taught, and the way in which a teacher reflects on the teaching experience
(Park & Oliver, 2008). This is because different topics may require different teaching
approaches, depending on the learning outcomes. This study sought to determine how PCK is
developed by investigating participating teachers through the use of multiple sources for data
triangulation.
In summary, Figure 2.1 describes the components of PCK that are likely to be used for
teaching statistics in school mathematics. They include subject matter content knowledge,
pedagogical knowledge, knowledge of learners’ conceptions and knowledge of learners’
learning difficulties. In the context of this study, the pedagogical knowledge in statistics
teaching will be assessed using multi-evaluation comprising of the lesson observation, written
reports, and questionnaire and documents analysis. PCK
Figure 2.1: Components of PCK used in this study
2.4.3
Pedagogical content knowledge and subject matter for teaching
Several researchers have used the terms ‘subject matter knowledge’ and ‘subject matter
content’ to describe the kind of knowledge that teachers need for teaching (Shulman, 1986;
Ma, 1999; Vistro-Yu, 2003; Jong, 2003; Jong et al., 2005; Halim et al., 2002; Rollnick et al.,
2008). In terms of mathematics teaching, Plotz (2007) referred to subject matter content
knowledge as ‘mathematical content knowledge’. With regard to PCK development in
statistics teaching it is necessary to define what each of the concepts means, so that they can
be used to define the construct of PCK that was used in statistics teaching. Plotz (2007)
argued that mathematical content knowledge is acquired mostly by studying mathematics in
31
school, and this may be described as ‘in-school acquired knowledge’. Van Driel et al. (1998),
Jong (2003) and Jong et al. (2005) described subject matter knowledge as the knowledge
obtained through formal training at universities and colleges, which may be regarded as
disciplinary education. From these assertions, it would seem that subject matter knowledge is
acquired through formal training in a subject area.
Ball and Bass (2000) researched the interweaving of content and pedagogy in the teaching
and learning of mathematics. The findings of their study indicated that the subject matter
knowledge needed by teachers is found not only in the list of topics of the subject matter to
be learned, but in the practice of teaching itself (Ball and Bass, 2000; Plotz, 2007). In other
words, knowing the content of a subject is not enough to justify the capacity of a teacher to
teach; what makes a teacher capable of teaching is also how well the teacher facilitates the
learning. According to these authors, little is known about the way in which ‘knowing’ a
topic from a list of topics affects teachers’ capabilities. And if one depends on analysing the
curriculum to identify the subject matter content knowledge needed for teaching the topics
without focusing on practice as well, not much will be gained (Ball and Bass, 2000; Plotz,
2007). Plotz’s (2007) study also reveals that mathematical content knowledge and
pedagogical knowledge are both needed for effective teaching and can motivate the
development of the PCK used for teaching. He stressed that teachers’ prior knowledge needs
to be exposed for effective content knowledge transformation and understanding as the prior
knowledge aided the teachers in the written problem-solving activities to design to assess
their mathematical content knowledge state.
Capraro, Capraro, Parker, Kulm and Raulerson (2005) researched the role of mathematics
content knowledge in developing pre-service teachers’ PCK, using performance in a previous
mathematics course, a pre- and post-test assessment instrument, success in the state-level
teacher certification examination, and journals. Their study outlined the connection between
mathematics content knowledge and pedagogical knowledge in developing PCK, in order to
address the increasing expectations of what learners should know and be able to do, and
knowledge that the teachers must have in order to meet the educational goals during
instruction and learning. A total of 193 undergraduate students who enrolled in integrated
method block courses prior to the teaching practice programme were involved in the research
project on teaching practice in mathematics. The findings of Capraro et al. (2005) indicated
that the teachers’ previous mathematics abilities are valuable predictors of students’ success
32
in teacher certificate examinations. Secondly, the mathematically competent pre-service
teachers exhibited progressively more PCK, as they had been exposed to mathematical
pedagogy comprising subject matter content and teaching practice during their mathematics
method course. Therefore, for one to have pedagogically powerful representations of a topic,
one should first have a comprehensive understanding of the topic.
Following this argument, subject matter knowledge in the context of PCK development
becomes a product of the interaction between mathematical competence and concern for the
instruction and learning of mathematics (Plotz, 2007). In other words, the concern for
instruction and learning shown by a competent mathematics teacher must demonstrate that he
or she has adequate knowledge of the subject matter, and this may be necessary for PCK
development. In this study, it is assumed that during their university preparation programmes,
the participating teachers acquired the subject matter knowledge of mathematics and the
pedagogical knowledge necessary for PCK development in statistics.
However, the South African mathematics (Grades 10–12) teaching force is made up mainly
of practitioners who have three-year teaching diplomas obtained from the old (pre-1994)
colleges of education (Rollnick et al., 2008). Less than 40% of these teachers hold a junior
degree on the subject they teach. The mathematics content measures only up to that of first
year at a university (Rollnick et al., 2008). In this study, the key question is, given that the
teachers show competence or understanding of these concepts in mathematics, irrespective of
their training, how does this influence their teaching and therefore their PCK for teaching
statistics in school mathematics?
Vistro-Yu (2003) conducted a study on how secondary school mathematics teachers faced the
challenges of teaching mathematics (in terms of the pedagogical knowledge requirements of
PCK in mathematics) in a new mathematics class in college algebra. Thirty-three secondary
school mathematics teachers were initially involved in the research project. They were made
to write a standardised test in high-school mathematics to determine the level of their subject
matter content knowledge. Based on this performance, six teachers were selected for the
research project. These six teachers were asked to prepare and teach an assigned topic in a
college algebra module, while the researcher conducted classroom observations of the lessons
presented by them. They were interviewed before teaching commenced, and after the lessons,
the six teachers were given a questionnaire to complete by reflecting on their teaching
33
performance. The findings of the study showed that the teachers were limited in the ways
they prepared their lessons. According to Vistro-Yu (2003), they were not able to teach in an
organised manner and lacked in-depth subject matter knowledge. The results of the interview
showed that some of the participants were dissatisfied with their teacher education
preparatory programmes because they lacked thorough content knowledge of the subject
matter. In this study, the methods adopted by Vistro-Yu (2003), namely teachers’ content
knowledge exercise, lesson observation, and interviews, were used to determine the subject
matter content knowledge and pedagogical content knowledge of these mathematics teachers
(the participants in the study).
Jong et al. (2005) conducted a study of the PCK of pre-service teachers using particle models
to teach chemistry at secondary-school level. Responses to written assignments, transcripts of
workshop discussions, and reflective reports by the participants were used to collect data. The
findings of this study indicated that the pre-service teachers were able to understand and
describe the learning difficulties of their learners during teaching with particle models. In
addition, they developed PCK using particle models, although development varied among the
participants (Jong et al, 2005).
The research methods of Ball and Bass (2000), Vistro-Yu (2003), Capraro et al. (2005) and
Jong et al. (2005) provided the rationale for the assessment of subject matter content,
pedagogical knowledge, knowledge of learners’ conceptions and learning difficulties as
constituent elements needed to develop PCK for teaching. However, there were deficiencies
in their studies. One of these was that their research was conducted within a relatively short
time (Vistro-Yu, 2003; Capraro et al., 2005). For instance, using one, two or four lesson
periods to conduct an investigation on the challenges in the instruction and learning of
mathematics (Vistro-Yu, 2003; Capraro et al., 2005) may not be adequate, since most topics
in mathematics take more than one period to teach.
Second, some of the researchers (Capraro et al., 2005; Ball & Bass, 2000) used grades
obtained in their university courses to justify the competency of a teacher in instructing a
subject. This may not be adequate, as the number of mathematics courses that a teacher has
studied at university or college does not necessarily ensure effective or quality teaching in a
classroom situation (Plotz, 2007; Capraro et al., 2005; Geddis, 1993). Rather, what makes
him or her an effective teacher is how well he or she understands what learners have to learn,
34
and the way he or she presents the subject matter content (Muijs & Reynolds, 2000; Graffin
et al., 1996). Therefore, more precise measures are needed to specify in greater detail the
relationships between the various components of PCK and how they are developed in order to
improve learner performance in mathematics (DoE, 2008). A third deficiency is lack of
lesson observation in conducting some of the investigations (Capraro et al., 2005; Ball &
Bass, 2000). The use of lesson observation would have afforded the researchers the
opportunity to determine how mathematics teachers use their PCK, for example preparation
and presentation of the lesson based on adequate knowledge of the subject matter; and
identification of learners’ preconceptions and learning difficulties, conceptions and
misconceptions concerning the topic (Jong, 2003).
In order to avoid these deficiencies, the study was carried out with the following features:
1) The PCK of teachers were investigated over a relatively long period (between four
and six weeks).
2)
The study was carried out with experienced secondary school mathematics teachers.
3)
Lesson observation was undertaken to determine how the teachers demonstrated their
PCK and subject matter knowledge during the teaching process and how they
identified learners’ preconceptions and misconceptions of the topic.
4)
Teachers’ and learners’ portfolios and workbooks were examined to determine what
had made the instruction and learning of the topic easy or difficult.
5)
These features were adapted to investigate the way competent mathematics teachers
developed their PCK for teaching statistics in school mathematics, in the hope of
discovering a further directive for the continuous improvement of the mathematics
teachers’ PCK in statistics teaching as well as of educational programmes for inservice and pre-service teachers of statistics.
In terms of measuring teachers’ subject matter content knowledge in a topic, several
techniques and methods have been used by several researchers in the field of mathematics
and science education. For instance, Gess-Newsome and Lederman (2001) and Jong (2003)
reported that a teachers’ subject matter content knowledge can be measured using concept
mapping, card sorting and pictorial representation. In this study, the subject matter content
knowledge of the participating teachers was assessed with the conceptual knowledge
exercise, concept mapping, interview and lesson observation.
35
The conceptual knowledge exercise in statistics was designed in multiple-choice formats. The
multiple-choice questions in statistics consist of a series of question, each with five possible
options from which the participating teachers have to choose the best to answer the questions.
Critics say that the multiple-choice format may not accurately depict the respondent’s
personal views about teaching because there is no provision for the reasons for the selection
of a particular option. But researchers continue to use multiple-choice questions with success,
because the many advantages of this type of question offset their demerits (Gess-Newsome
and Lederman, 2001; Kazeni, 2006). For example, multiple-choice questions can be set at
different cognitive levels. They are versatile if designed and used appropriately (Miller,
2006). Multiple-choice question assessments can be completed in a short time, and they
ensure better coverage of content. In this study, multiple-choice questions were used to assess
the changes in statistics content knowledge of the participants (since they have been teaching
the topic) as they may have covered enough content area of statistics and to select them for
the second phase of the qualitative research.
Considering the role of concept mapping in teaching and learning, Ochonogor and Awaji
(2005) and Novak and Cannas (2006) described concept mapping as a learning strategy that
aids understanding of complex ideas and clarifies ambiguous relationships between ideas.
According to these authors, concept maps may be seen as graphical tools for representing
topics, by depicting key concepts and organising knowledge clearly. Following this
argument, organising and representing the knowledge of a particular topic can take the form
of connecting the concepts by means of arrows, boxes, words or phrases in order to elicit the
meaning of the relationships between the concepts. In this connection, concept maps are seen
as a special form of web diagram for exploring knowledge and gathering and sharing
information visually (Novak & Cannas, 2006). Concept maps can depict how we think, which
influences how and what we teach (Miller, 2006). Hence, concept maps can provide
opportunities to see relationships between types of knowledge.
Novak and Gowin (1994:96) argued ‘that concept maps provide visual representations of
knowledge’. According to these authors, concept maps allow researchers to create concrete
representations of knowledge that can be used to determine knowledge changes in a teacher.
Since concept maps create physical representation of knowledge, changes in this
representation are assumed to provide evidence of teacher knowledge change (Miller,
2006:96).
36
Miller (2006) used concept maps to analyse the construction of pre-service teachers’ PCK
during a science method course. The participants of the study were asked to construct a
concept map of important concepts in a specific chemistry unit that focuses on numerous
teaching activities. The findings of this study show that the changes in the structure of the
concept map were related to the changes in the personal knowledge of the learner.
Ferry, Hedberg and Harper (1997) investigated how pre-service teachers used a concept map
to organise curriculum content knowledge. Participants of the study were asked to use a
concept map to plan science-based instruction that could be delivered to an elementary
science class. The results of the study showed that pre-service teachers had different
perceptions of the connections between the basic statistical concepts, which enhanced their
conceptual understanding of the concepts and aided the sequential planning of the sequence
of the concepts for teaching (Ferry, Hedberg & Harper, 1997).
Concept mapping may lack reliability in terms of representing all that an individual knows
about the content knowledge being assessed (Miller, 2006). Furthermore, if a teacher does not
continue with classroom practice, the changes in knowledge of the topic may be short lived.
However, concept maps have been credited with many advantages. For instance, a concept
map allows teachers to organise their knowledge of teaching their primary content area much
better with high cognitive demand. In this study, a concept mapping exercise was used to
indirectly assess teachers’ content knowledge of statistics in school mathematics by arranging
statistics topics in logical sequence according to the way in which the teachers would present
them in their classroom practice.
The interview was used to triangulate the data gathered with the concept mapping. The
interview consists of open-ended questions that the interviewer asked the interviewees to
respond to. The interview allows the respondent the opportunities to create options for
responding and to voice their experiences unconstrained by any perspective of the researcher
or past research that may not directly be observed in the respondent action (Cresswell,
2008:225). Some researchers argued that an interview is deceptive and provides the
perspective the interviewees want the interviewer to hear, which renders the information
inarticulate, perceptive and unclear (Cresswell, 2008). Several researchers (Vistro-Yu, 2003;
37
Loughran et al, 2004; Hill, 2008) have used the interview to assess teachers’ educational
background that must have assisted them to develop their topic-specific content knowledge
and PCK. In this study, an interview schedule was used to gather data to assess the teachers’
educational background that had enabled them to develop their topic-specific content and
PCK in statistics teaching. The use of lesson observations in assessing teachers’ content and
pedagogical knowledge will be discussed in Section 2.4.4.
The research procedures used by researchers such as Jong et al. (2005), Capraro et al. (2005),
Vistro-Yu (2003), Jong (2003) and Van Driel et al. (1998) share the same research procedure
as this study in terms of the use of these instruments: a conceptual knowledge exercise,
interview schedules, concept mapping, to assess subject matter content knowledge and PCK.
2.4.4
PCK and pedagogical knowledge (instructional skills and strategies)
Pedagogical knowledge is believed to be the kind of information that a teacher needs and
uses to perform everyday teaching tasks. It involves teaching styles and strategies, classroom
management and teaching and learning processes relating to learners in the classroom
(Cochram et al., 1993; Vistro-Yu, 2003). Pedagogical knowledge includes knowing and
understanding the content to be taught and the specific demands of that content, such as
instructional skill and strategies (Kreber, 2004; Loughran et al., 2004; Ball, Thames &
Phelps, 2008). Instructional knowledge entails knowing how to sequence the learning
outcomes, prepare the lessons, facilitate discussion and group work, construct tests and
evaluate learners’ understanding through the use of examinations, among others (Kreber,
2004).
In general, different kinds of instructional strategies, representations and activities are used in
teaching mathematics. Knowledge of instructional strategies entails understanding ways of
representing specific concepts, in order to facilitate student learning. Representations include
illustrations, examples, models, and analogies. Each representation has a conceptual
advantage and disadvantage over other representations (Ibeawuchi, 2010). PCK in this area
includes awareness of the relative strengths and weaknesses of a particular representation.
Activities can be used to help learners understand specific concepts or relationships, for
example demonstrations, simulations, investigations and even experimentations. PCK of this
type incorporates teachers’ knowledge of the conceptual power of a particular activity
38
(Magnusson et al., 1999). For a representation to be powerful or comprehensible, the teacher
must know the learners’ conceptions about a particular topic, and the possible difficulties
they will experience during the teaching and learning of the topic. Representations during
teaching must be clearly linked, and the relationships between concepts must be
comprehensible (Ibeawuchi, 2010). However, most mathematics teachers are not able to
identify learner misconceptions and to teach for conceptual change since most of them have
not yet dealt with their own alternative conceptions, and are working with very limited
resources, time, and necessary skills (Van Driel, 1998).
Several studies have highlighted certain instructional strategies as a component of PCK.
Hashweh (1987) for example emphasises that incorrect and misleading representations, such
as analogies and examples that depict the teachers’ misconceptions, could result from
teaching outside one’s own field of expertise. Tobin, Tippins and Gallard (1994) also state
that when teachers teach outside their areas of specialisation, they give explanations and
analogies that reinforce the misconceptions that learners already have.
Magnusson et al. (1999) argue that pedagogical knowledge as a component of PCK is
dependent on teachers’ subject matter knowledge about a particular concept. This may not
always be true, as subject matter knowledge does not guarantee that PCK will be transformed
into representations that will help learners understand targeted concepts, or that teachers will
be able to decide when it is most appropriate pedagogically to use a particular representation.
Anderson and Mitchener (1994), in their research on science education, support this view and
are of the opinion that teachers’ knowledge of science teaching may be limited, even if the
teachers have knowledge of the subject matter. In a particular topic, pedagogical knowledge,
or the way concepts are represented as a component of PCK, seems to depend on previous
planning, teaching, and reflection (Halim & Meerah, 2002).
Vistro-Yu (2003) researched pedagogical knowledge in mathematics and focused his study
on how the mathematics teacher faces the challenge of teaching algebra in a new class. As
explained earlier, pedagogical knowledge is knowledge used for teaching, particularly
awareness of instructional techniques, psychological principles, classroom management, and
the teaching and learning process. Similar PCK-related studies by Jong et al. (2005) and
Rollnick et al. (2008) show that science teachers with adequate pedagogical knowledge
should be able to design good teaching and learning strategies that allow them to teach the
39
concepts and manage the classroom and other instruction and learning processes. Hence, the
instructional strategies used by the participants in the study for teaching school statistics were
investigated in classroom practice. The question that one would ask at this stage is how do we
measure the knowledge of instructional skills and strategies demonstrated by the teachers in
their statistics lesson.
Current researches on PCK have suggested that the multi-method approach may be
appropriate in exploring knowledge of the relevant instructional strategies (Jong, 2003;
Miller, 2006; Rollnick et al., 2008; Ibeawuchi, 2010; Toerien, 2011) during classroom
practice. Multi-method evaluation involves collecting multiple sources of data. Multi-method
analysis tends to create increasing impact on changing knowledge, with each data source
adding more dimensions to the findings from another source, thereby biasing the findings of
the study (Gess-Newsome & Lederman, 2001). Nevertheless, researchers are using this
method with increasing success. Multi-method evaluation is useful for triangulation of data
and improving the validity of the data (Gess-Newsome & Lederman, 2001). In this study,
multiple sources were used to collect data to assess the instructional skills and strategies that
the participating teachers used in teaching statistics.
One of the multiple sources is the lesson observation of the participating teachers. Lesson
observation is a process of gathering open-ended, firsthand information by observing the
participant physically and gathering the information as it occurs at the research site
(Cresswell, 2008:221). Lesson observation has the advantage of studying the actual
behaviour of the participants and the difficulties they may have in demonstrating their ideas
during research activities. The disadvantages of using lesson observation for data collection
are that the researcher will be limited to the site and situations of the research and may have
difficulty in establishing rapport with individuals. But despite the disadvantages, researchers
continue to use lesson observation with success because of the firsthand information and
recording the actual behaviour of the participants at the research site. The lesson observation
was also used to triangulate data gathered with the concept mapping exercise (ref Section
2.4.3).
In this study, the teachers’ written reports were triangulated with learners' lesson observations
which form part of the multiple sources for evaluating teachers pedagogical knowledge in
statistics teaching. Several researchers, including Gess-Newsome & Lederman (2001), Penso
40
(2002) and Jong (2003), Capraro et al (2005), have used the teacher’ written report to
evaluate teachers’ PCK during classroom practices in science and mathematics. It has the
advantage of making teachers reflect on their teaching, thereby providing opportunities for
the teachers to evaluate it. In this study, the teachers’ written reports were used to assess the
teachers’ pedagogical and triangulate the data collected with lesson observation in terms of
reflecting on what transpired during the lesson.
Researchers such as Gess-Newsome and Lederman (2001) and Vistro-Yu (2003) have used
questionnaire to determine teachers’ pedagogical knowledge in the context of PCK
development. According to them, they were able to capture what the teachers did while
teaching a specific topic in science and mathematics. In this study, part of the teacher
questionnaire responses was used to assess what the teachers did while teaching the assigned
topic in statistics. Free-response questionnaire allows the researcher to obtain the teachers’
feelings about their actions during the lesson, which they might not have displayed or
expressed during the lesson and interview.
The documents analysis and video records were also used to triangulate the data from the
lesson observation. Capraro et al (2005), Jong et al (2005) and Ogbonnaya (2011) have used
document analysis such as journal and certification to gather data to assess the teachers’
content and pedagogical knowledge in mathematics and they were successful in gathering
data related to the teachers’ content and pedagogical knowledge. In this study, the documents
analyse included the teacher portfolios, learners’ workbook and portfolios, textbooks as well
as school policy guidelines for teaching and learning. They have the advantage of being
readily available for reading, analysis and interpretation to the researcher.
Based on these advantages, the documents (learners class workbooks and portfolios, teacher
portfolios, lesson plans, and NCS subject assessment guidelines) were considered as a source
for gathering data to assess the teachers’ pedagogical knowledge in terms of what has made
the lesson easy or difficult
Jong (2003:375) explained that teachers are able to explain their cognition in detail while
they look at a video record of a lesson that has been taught. Because of the distracting effect
of a video recording being made in the classroom, an interview can be considered a
replacement for it. The video recording is used as a tool for teachers to remember what they
taught during the lesson, and they can experience how the lesson was delivered, unlike the
41
interview, which only allows the respondents to verbalise their actions during the lesson.
Jong (2003) noted that the stimulated-recall interview (video records) might be more
appropriate in explaining teachers’ actions during classroom practice. In this study, the video
recorder was used to record the lessons in which the participating teachers demonstrated their
pedagogical knowledge in statistics teaching and to triangulate the lesson observations in
statistical graphs.
2.4.5
PCK and knowledge of learners’ preconceptions and learning difficulties
Instructional strategies, learning difficulties and misconceptions are some of the components
of pedagogical content knowledge that are used in teaching a particular topic in a specific
subject area (Penso, 2002). Penso (2002) conducted a study on the PCK of pre-service
biology teachers, with the emphasis on how student teachers identify and describe learners’
learning difficulties. The teacher used classroom observation and learners’ diaries to collect
data from the participants. Penso’s (2002) findings showed that learning difficulties could be
identified and described during teaching and by observing lessons. Penso (2002) claimed that
these difficulties might originate from the way the lessons were taught, which involves the
content of the lesson, lesson preparation and implementation, and the learning atmosphere.
Other factors include the misconceptions that the learners and the teachers have about the
topic, and the cognitive and affective characteristics of the learners.
According to Penso (2002), learners regard their learning difficulties as being caused by
conditions prior to the process of teaching and to those existing in the course of teaching.
While the aspect of lesson content relates to the level of difficulty and abstraction of the
topic, the teaching, lesson preparation and implementation aspects are concerned with the
structure and presentation of the lesson (Cazorla, 2006). Negative lesson structure conditions
include overloading content and unsatisfactory sequences in the lesson. Negative lesson
presentation conditions include inappropriate instructional strategies for presentation, and not
contributing to the process of learning. Negative cognitive and affective characteristics entail
lack of prior knowledge about a topic that would enable learners to cope with the lesson in a
meaningful way, preconceptions developed by the learners because of previous experiences,
partial and inconsistent thinking, and lack of motivation and concentration. These negative
cognitive and affective characteristics may result in learning difficulties in a teaching and
learning situation if the teacher does not have adequate prior content knowledge of the topic.
42
Cazorla (2006) researched the ways in which mathematics teachers teach statistics in
elementary and secondary schools and teacher training colleges, and reported that
mathematics teachers seemed to encounter teaching and learning difficulties during teaching.
According to this author, misconceptions and the ways in which mathematics lessons are
taught are among the factors that contribute to learners’ learning difficulties in statistics
teaching. In addition, most statistics teachers do not have adequate knowledge of the
curriculum and the necessary approaches to the teaching and learning of statistics. This leads
to poor content delivery in the classroom, and consequently affects learners’ performance.
Jong (2003), in his research on exploring science teachers’ pedagogical content knowledge,
used a teacher’s log, concept mapping, interviews, and convergent and inferential
investigation techniques and notes in order to identify and resolve misconceptions and
learning difficulties. Convergent and inferential techniques may be used by the teachers
during classroom practice. These refer to data collection techniques in which questions are
developed in short-answer and multiple-choice formats to probe the preconceptions and
misconceptions of learners in a topic (Jong, 2003). The gap in this study is that lesson
observation could have been used to determine how teachers use their PCK to identify
learning difficulties during the lesson.
It is thus conclusive that inadequate subject matter knowledge and inappropriate instructional
strategies employed in classroom practice can bring about misconceptions and learning
difficulties among learners in statistics teaching. However, learning difficulties can be
resolved if practising teachers have developed adequate PCK to solve them, which, in turn,
can lead to improved learner achievement. In this study, the teachers’ knowledge of learners’
learning difficulties was assessed through lesson observation, questionnaires, teachers’
written reports and document analysis.
In the literature review, the studies by Penso (2002) and researchers such as Jong et al.
(2005), Jong (2003), Van Driel et al. (1998), Capraro et al. (2005) and Cazorla (2006) justify
the need for this study to investigate how competent secondary school mathematics teachers
develop PCK in statistics teaching.
43
Research reports by Jong (2003) and Gess-Newsome and Lederman (2001) indicated that
convergent and inferential techniques may be appropriate in measuring teachers’ knowledge
of learners’ preconceptions and learning difficulties in science. The convergent and
inferential technique involves the use of predetermined verbal descriptions of teacher
knowledge comprising multiple choices and short-answer questionnaire. A multiple-choice
item test is a series of questions with several possible answers, from which a person has to
choose the correct one. The multiple-choice format can be used to rate individual
performance and ability in a test, as well as to compare the performance between participants
(as in this study) (Bontis, Hardie & Serenko, 2009; Kehoe, 1995).
In this study, the teachers’ knowledge of learners’ preconceptions and learning difficulties
were assessed using the lesson observation, as part of the interview schedule, and in the
questionnaire, written reports and documents analysis. Based on the way various researchers
used these instruments in assessing teachers’ content and pedagogical knowledge, and the
many advantages of using them to capture teachers’ PCK (ref Sections 2.4.3 and 2.4.4), the
lesson observation was adapted to assess the teachers’ knowledge of learners’ preconceptions
and learning difficulties in statistics teaching in order to attest how this knowledge manifests
in the teacher during classroom practice. The data gathered with the interview, questionnaire,
written reports and documents analysis were used to triangulate the lesson observation and to
ascertain how the teachers’ knowledge of learners’ preconception and learning difficulties
manifests during the lesson on statistical graph.
2.5
Summary of the chapter
In this chapter, various categories of relevant literature on PCK were presented. It began with
a description of the NCS for Mathematics and Statistics, and explained how these subjects
relate to each other. Although the studies of Penso (2001), Gess-Newsome and Lederman (2001),
Rollnick et al (2008) and Jong (2003) were in the area of the sciences, their framework for describing
the PCK in science teaching seemed relevant to describing how the participating teachers developed
their PCK in statistics teaching. The researches on teaching and learning statistics, mathematics
and sciences provide the benchmarks and suggestions about the process that the study has to
consider in describing how the participating teachers develop PCK in statistics teaching. PCK
is an appropriate theoretical framework for the study as it addresses the key issues: subject
matter content knowledge, pedagogical knowledge, knowledge of learners’ conceptions and
knowledge of learners’ learning difficulties, and bridging the gap in PCK development in
44
statistics teaching. The chapter concluded with a detailed description of how the components
of PCK used for this study were assessed to determine the individual topic-specific PCK in
statistics teaching.
45
CHAPTER 3
3.0
RESEARCH METHODOLOGY AND PROCEDURE
3.1
Introduction
This chapter discusses the research methodology and procedures adopted for collecting data.
It starts with a description of the research design, followed by the research method, and ends
with an outline of the statistical techniques used to address issues of validity and reliability of
the instruments used for the collection of data.
3.2
Assumption of PCK development during classroom practice
It was assumed that competent mathematics teachers would have developed their PCK, which
enables them, through classroom teaching, to improve learners’ performances at the Senior
Certificate Examination over time. Observing the participating teachers prepare and teach a
lesson in an assigned topic would enable the researcher to determine how they developed
their topic-specific PCK in statistics teaching.
3.3
3.3.1
Research design and method used in this study
Research design
The study adopted a descriptive research design using the case study research method.
Descriptive research investigates and describes a case about the current situation of an event
or how it has happen in the past (Mayer & Fantz, 2004). It is used to tease out possible
antecedents of an event that happened in the past. It is assumed that the competent
mathematics teachers have developed adequate PCK, which enables them to improve their
learners’ performance in the Senior Certificate Examinations over time. A descriptive
research design was considered appropriate for the nature of the topic under investigation
because this study intends to investigate how the teachers developed their PCK over time.
3.3.2
Research method
This study used a qualitative research approach utilising a case study method. Creswell
(2008) defines the case study method as ‘an empirical inquiry that investigates a
contemporary phenomenon within its real-life context when the boundaries between
phenomenon and context are not clearly evident and when multiple sources of evidence are
46
used’. This study sought to investigate how competent mathematics teachers developed their
PCK in teaching statistics in their statistics lesson. There is some criticism of the use of case study research methods. ‘Critics believe that a
small number of cases cannot offer adequate grounds for establishing reliability or generality
of findings’ (Yin, 1984). Others feel that intense exposure to the study of a case biases the
findings (Yin, 1993 & 1994; Feagin, Orum & Sjoberg, 1991). Some argue that case study
research is useful only as an exploratory tool (De Vos, 2000). However, researchers continue
to use the method successfully in carefully planned practical studies of real-life situations,
issues and problems (Soy, 2006). Soy (2006) argued that successful use of case studies in
conducting investigations in scientific studies, despite the criticisms, has many benefits, such
as providing a rich and detailed account of the case in a real-life context. The case study was
chosen for this research in order to provide a rich and detailed account in a real-life context of
how the mathematics teachers develop their PCK in statistics teaching. It is considered
adequate and conventional in the field of the author’s research interest, as it is used to collect
information in order to gain greater insight into and understanding of the way in which PCK
may have been developed by competent teachers. This study is a qualitative one that uses both quantitative and qualitative data. The
quantitative data was gathered through the conceptual knowledge exercise for teachers and
concept mapping. The participants’ performance in these exercises involved their marks
(expressed in percentages). Interview schedules, observations of lessons, teacher
questionnaires, teachers’ written reports, video recordings, and document analysis were used
to collect qualitative data. The individual teacher’s PCK and its development in data handling
teaching/statistics constituted the unit of analysis in this study.
3.4
3.4.1
Population and sample description
Study population
The population of the study comprised Grade 11 mathematics teachers in Tshwane North
District, Gauteng, South Africa. There are twelve high schools in Tshwane North District.
With a criterion of 70% for learners’ performance in the Senior Certificate Examination in
Mathematics for a period of two years, seven schools were identified from which the
participating teachers were selected. The identification of the schools was followed by
47
interviews with the principals, peers and subject specialists at the Department of Basic
Education (DoBE) to identify the willing participating teachers.
3.4.2
Study sample
The teachers in the main study were selected, through a process of elimination, according to
certain criteria: learners’ performance in mathematics in the Senior Certificate Examination;
recommendations by school principals, subject specialists at the Department of Education and
peers; and competence in statistics through performance in a statistics test. Tshwane North
Education District Cluster 3, Gauteng Province, comprises twelve schools. Of these schools,
only seven had scored a minimum of 70% mathematics pass rate for two consecutive years in
the Senior Certificate Examination. Mathematics teachers from these schools were invited to
volunteer for the project. Six teachers from six separate schools indicated their willingness to
participate. The researcher requested recommendations from principals, peers and subject
specialists from the Department of Basic Education (DoBE) for these teachers. Based on their
recommendations, six teachers were selected. Finally, the six teachers wrote the conceptual
knowledge exercise in statistics. The top four scorers were selected for the main study. Table
3.1 summarises their performances, and their demographic profiles are described in section
4.3.
Table 3.1:
Schools and teachers that participated in the main study
S/NO
SCHOOL
NSC RESULTS
TEACHER
1
School A
81%
Teacher A
2
School B
94%
Teacher B
3
School D
93%
Teacher D
4
School E
98%
Teacher E
3.5
3.5.1
3.5.1.1
Research instrument used for collecting data
Development of research instruments
Teacher conceptual knowledge exercise in statistics
The conceptual knowledge exercise was adopted to collect data in this study.
48
The National Curriculum Statement for Mathematics for the Senior Phase of the Further
Education Training (FET) bands for Grades 10–12 and the prescribed textbooks were
reviewed and analysed. The aim was to ascertain the targeted knowledge, competence and
skills for developing the test items based on the mathematics assessment taxonomy. A large
number of multiple-choice test items were initially formulated by the researcher from sources
such as public examinations, locally prepared past examinations and tests, selection tests,
achievement tests and textbooks in mathematics. The items were designed in line with
Bloom’s Taxonomy and the South African Mathematics Assessment Taxonomy, as indicated
in the examination guidelines of the NCS (DoE, 2008) and Table 3.2. The competencies
tested according to Bloom’s Taxonomy included knowledge, comprehension, analysis,
synthesis, application and evaluation (DoE, 2010). The levels of the mathematics assessment
taxonomy are knowledge (level 1); applying routine procedures in familiar contexts (level 2);
applying multi-step procedures in a variety of contexts (level 3); and reasoning and reflecting
(level 4) (DoE, 2010). Comprehension and application of Bloom’s Taxonomy were used to
design the conceptual knowledge exercise, in line with the mathematics assessment
taxonomy. The mark allocation was the total mark allocated to all items that were developed
according to levels. For instance, all marks allocated to level 1 questions that test knowledge
in any mathematics test or examination must not exceed 20 out of the total mark of 100 for
the examination or test.
Table of specification 3.2:
LEVELS OF
ASSESSMENT
Mathematics assessment taxonomy and marks allocation
ASSESSMENT TAXONOMY
MARKS
ALLOCATION
1
Knowledge
20
2
Applying routine procedures in familiar contexts
25
3
Applying multi-step procedures in a variety of contexts
30
4
Reasoning and reflecting
25
(DoE, 2010)
49
Table of specification 3.3:
COMPETENCE
Showing competency and skills and marks allocated
ABILITIES
Comprehension Applying routine
(understanding) procedures in
familiar contexts
Applications
Applying what
was learnt in the
classroom in
solving problems
in familiar or other
situations by using
routine, multi-step
procedures
SKILLS
DEMONSTRATED
Grasping
(understanding) the
meaning of
informational
concept/materials
Solving problems using
required skills or
knowledge
QUESTION
MARKS
TOTAL
ALLOCATED
1, 2, 3, 6, 11,
13, 15, 20
5 for each
item
40
4, 5, 7, 8. 9,
10, 12, 14 16,
17, 18, 19
5 for each
item
60
TOTAL
100
(DoE, 2010)
The conceptual knowledge exercise included 40% of the questions designed to test
comprehension and consisted of level 2 and 3 questions in statistics (ref Table of
specification 3.2). Examples of items measuring comprehension knowledge are 1, 2, 3, 6, 11,
13, 15 and 20 (ref Table of specification 3.3). Below is an example of the levels 2 and 3
questions.
Use the frequency distribution table below to answer question 2
2
Interval
0-4
5-9
10-14
15-19
20-24
Frequency
3
5
7
4
1
Estimate the mode of the distribution
The remaining 60% tested application knowledge at levels 3 and 4, where participants had to
apply higher-order thinking to solve problems in statistics (ref Table of specification 3.2).
Examples of items measuring application of knowledge are 4, 5, 7, 8, 9, 10, 12, 14, 16, 17, 18
and 19. The question below is an example of levels 3 and 4 questions.
4
The mean height of three groups of students consisting of 20, 16 and 14 students is 1.67m,
1.50m and 1.40m respectively. Find the mean height of all the students.
50
The conceptual knowledge test was designed to determine how well the teachers could
demonstrate that they had adequate content knowledge of the topic by applying routine and
multi-step procedures, as well as reasoning and reflection. Initially 30 multiple-choice test
items were developed in statistics from the sources indicated, each with five possible
responses. Only one of the five options was correct. These items were scrutinised by
mathematics experts at the DoBE, and national examiners in NCS mathematics (ref Appendix
XXII). The responses from the reviewers were used to modify the test items that formed the
first draft of the instrument. For example, item 4 asked, ‘The mean heights of three groups of
students consisting of 20, 16 and 14 students are 1.67 m, 1.50 m and 1.40 m respectively.
What is the mean height of all the students?’ The item was modified to ‘Find’ instead of
‘What’, as previously used in the question.
•
Scoring the test items One mark was allocated to each item. The total mark for the 20 items was therefore 20 marks.
While the comprehension part of the question was 8 marks, the application part was 12
marks. For the correct answer to each question, one mark was awarded in both the
comprehension and application parts of the question. The marks were later converted to 100
marks. Selection of participants for the concept map and qualitative aspect of the research
was based on performance in the conceptual knowledge exercise. A teacher had to score a
minimum of 70% to be adjudged to have adequate subject matter content knowledge of
statistics in school mathematics.
3.5.1.2
Concept mapping for teachers
The NCS was used to compile the list of contents of statistics in school mathematics. The
topics according to the NCS for Grades 10 to 12 are stem-and-leaf; mode, median and mean
of ungrouped data; frequency table of grouped data; range, percentiles, quartiles; interquartiles and semi-quartile range; bar and compound bar graphs; histograms; frequency
polygons; pie charts; line and broken line graphs; box-and-whisker plots; variance, mean
deviation; standard deviation; ogives; five number summaries; scatter plots; lines of best fit
(DoE, 2010) (ref Appendix XXIV).
The participating teachers were required to use the topics listed above to construct a concept
map. The question states:
51
(a)
Arrange the topics in each grade on how you think they should be taught in grades 10, 11 and
12.
(b)
With an arrow, show how you can teach these topics sequentially in each grade. For example,
you observe morning before afternoon and before evening. Therefore;
Morning
afternoon
evening
For example, in measures of central tendency, the mode is taught first, followed by the
median and the mean. Therefore, the memorandum for question (a) should be:
Table 3.4:
Table showing the list of statistics taught in grades 10, 11 and 12 (if any)
GRADE 10
GRADE 11
GRADE 12
Mode, median, mean, ranges,
(ungrouped data), frequency
table, bar and compound bar
graphs, histogram, frequency
polygons, pie charts, line and
broken line graphs. mode,
median and mean (grouped
data), quartiles, inter-quartiles
and semi-inter-quartile range
Five number summary, box and
whisker diagrams, ogives,
variance and standard
deviation, scatter diagrams,
lines of best fit
N/A
An example of how question (b) should be answered for grade 10 is:
Mode
Median
Frequency table Bar
Polygon
Mean Ranges
(Ungrouped data)
and Compound bar graphs
Frequency
Pie Charts Line and broken line graphs.
Mode
Median
Quartiles
Inter-quartile and semi-inter-quartile ranges
•
Histogram
Mean (Grouped data)
Scoring of concept mapping
A rubric was designed by the researcher to indicate how to evaluate the concept map drawn
by the participants. It allocated marks to the number of topics that were correctly arranged,
and deducted marks for incorrect arrangement of topics (ref Appendix XXV). As indicated in Appendix XXV, marks were allocated for the number of topics that were
correctly arranged, and deducted for incorrect arrangement of topics in each grade. The mark
52
allocation for the concept mapping exercise was 25 marks in each grade for question a). The
combined mark for Grades 10 and 11 for question a) was 50. Question a) requested the
participating teachers to ‘Arrange the topics in each grade on how best they can be taught in
Grades 10, 11 and 12’. No mark was allocated for Grade 12, as the topic is not taught in that
grade. The same scoring system was applied to question b), in which the participants were
requested to ‘With an arrow, show how you can teach these topics sequentially in each grade.
For example, you observe morning before afternoon and before evening. Therefore, in a
sequential order, it is
Morning
Afternoon
Evening. A teacher who scored less than 60 marks could be
regarded as not having the knowledge of the curriculum that would inform his or her insight
into the topic. The reason for allocating the same mark is that each question required
approximately the same time to solve.
3.5.1.3
Interview schedule for teachers
The purpose of the semi-structured interview was to gain some insight into mathematics
teachers’ content knowledge and educational background that may have enabled them to
develop their topic-specific PCK in statistics. The semi-structured interview schedule was
based on several literature sources on PCK (e.g. Jong, 2003; Jong et al., 2005; Van Driel et
al., 1998; Rollnick et al., 2008)). To this end, questions were developed to address the
teachers’ teaching experience, qualifications, educational background and professional
development, knowledge of instructional strategies, and preconceptions in teaching and
learning statistics. The questions were grouped according to the components of PCK being
assessed in this study. This approach has been used by several researchers (Jong, 2003; Jong
et al., 2005; Van Driel et al., 1998; Rollnick et al., 2008) in the fields of mathematics and
science education. The distribution of the questions is shown in Table 3.5. The questions are
indicated in Appendix XXVI.
53
Table of specification 3.5:
Item specification table for the interview
PCK
components
Subject matter content
knowledge
Instructional skills and
strategies
Learning
difficulties
Workshop
Number of
items
1–9
10–13
14
15–20
Questions 1 to 9 were used to assess the teachers’ subject matter content knowledge and
demographic profile in statistics teaching. For example, question 1 asked:
‘What university/college did you attend?’
They were then asked to indicate the course they had studied in their disciplinary education
programme and their understanding of the nature of statistics in school mathematics.
Questions 10 to 13 probed the instructional strategies that they used for teaching statistics and
why they employed these strategies. For example, in question 12, participants were asked:
‘If the learners have any problem in understanding the topic based on the
instructional approach, what do you do to help them to understand?’
Question 14 was used to determine the learning difficulties that teachers themselves think
learners have about the topic. For example, the teachers were asked:
‘What learning difficulty do you remember experiencing as a pupil and as a university
student or from teaching experience in statistics?’
Questions 15 to 20 focused on workshops that the teachers had attended. For instance, the
teachers were asked:
‘Have you ever been to a mathematics workshop or teacher development
programme?’
The data related to workshops were used to triangulate data on teachers’ content knowledge.
54
Prior to the validation of the teacher structured interview schedule, it was given to three
secondary school Grade 11–12 mathematics teachers for comments about the categories and
educational background for developing PCK. Their comments were used to review the
questions before the pilot study.
3.5.1.4
Lesson observation schedules
The lesson observation schedules (ref Appendix XXIX) were standard ones recommended by
the Provincial Department of Education for normal classroom practice (DoE, 2010). The
schedule was therefore adopted for gathering data for assessing instructional knowledge used
in teaching statistics, which is the major focus of this study. The purpose of using the
standard lesson observation schedule was to collect data from real-life situations and to assess
how well the teachers prepared for lessons, as well as to check for consistency in their
implementation of plans (Vistro-Yu, 2003 & DoE, 2010) (ref Appendix XXIX).
3.5.1.5
Teacher questionnaire
The teacher questionnaire was designed to assess teachers’ PCK in terms of their knowledge
of instructional skills and strategies, learners’ conceptions in teaching and learning statistics,
and learning difficulties. The teacher questionnaire (ref Appendix XXVIII) consisted of 16
questions designed to triangulate data collected during lesson observation. Questions 1 to 9,
12, 13, 15, and 16 were used to assess the instructional strategies that the teachers used in
classroom practices in statistics teaching. An example of the questions focusing on
instructional skills and strategies is:
How did you identify the prior knowledge (preconceptions) which the learners bring to the
class about statistical graphs?
Questions 10, 11 and 14 were used to determine the learning difficulties that learners have
with the topics in statistics teaching (ref Table 3.6) (ref Appendix XXVIII). An example of
the questions is:
What is it about statistics that makes the learning easy or difficult?
55
Table of specification 3.6:
Item specification table for the questionnaire
PCK components
Instructional skills and strategies
Learning difficulties
Number of items
1–9, 12, 13, 15, 16
10, 11, 14
The questionnaire focused on what the teachers actually does while teaching, namely their
strategies or approach and methods (items 7–11, 15–16) and contents of the lessons (item 2).
Other information related to how the teacher identified learners' preconceptions and learning
difficulties (items 4–6, 10, 17), how these difficulties were resolved (items11, 12, 14), and
how the lessons were evaluated (items 13, 15 and 16) (ref Appendix XXVIII). As regards
teachers’ instructional strategies and skills, participants were requested to indicate the
duration of the lesson, topic, and essential prior knowledge (ref Appendix XXVIII). In
addition, participants were requested to indicate how learners responded to the class
activities, homework and assignments (ref Appendix XXVIII). For instance, the teachers
were asked, ‘How did learners respond to class activities, homework and assignments?’
Knowledge of learners’ conceptions and learning difficulties was assessed by asking the
teachers to indicate how they identified learners’ preconceptions and misconceptions, if any,
as well as learning difficulties in the context of teaching (ref Appendix XXVIII). For
example, the participating teachers were asked, ‘How did you identify the prior knowledge
(preconceptions) that the learners bring to the class about statistical graphs?’ Table 3.6
displays how the questions were distributed according to the various components of PCK,
namely instructional strategies and learning difficulties, and how the components were
assessed. The questionnaire was administered to the participants immediately after the last
lesson had been observed. 3.5.1.6
Teacher written reports
The teachers’ structured written reports (ref Appendix XXVII), in which they recorded what
made the lessons easy or difficult, were used to assess instructional strategies and learners’
learning difficulties after a four-week period of teaching statistics. The purpose of the
teachers’ written reports was to determine what (for the teacher) made the lessons easy or
difficult, and to triangulate other data related to how the teachers developed their PCK over
time. The written reports were compiled from teachers’ and learners’ portfolios, as well as
56
learners’ workbooks. For instance, the participating teachers were asked, ‘How did learners
respond to classroom activities as well as homework or assignments?’ The teachers’ portfolios contained information such as a formal programme of assessment in
mathematics for Grade 11, mathematics assessment tasks (standardised tests, assignments,
investigations or projects and examination papers), tools for assessments (memoranda,
checklists, rubrics, etc), and model answers for all assessment tasks. The learners’ portfolios
contained continuous moderation reports, a summary of marks, tests, examinations, and
assessments (DoE, 2010).
Table of specification 3.7:
Item specification table for the written reports
PCK components
Instructional skills and strategies
Learning difficulties
Number of items
5 and 6
1–4 and 7-9
Nine questions were formulated as guidelines for the teachers in compiling the report.
Questions 1 to 4 and 7 to 9 were used to examine learning difficulties, and questions 5 to 6
were used to determine instructional skills and strategies (ref Appendix XXVII). An example
of an item focusing on learning difficulties is:
What learning difficulties do you identify in learners when teaching statistical graphs?
An example of questions focusing on instructional skills and strategies is:
How did the learners respond to classroom activities as well as homework or
assignments?
The reports were given to experienced mathematics teachers in Grades 11 and 12, who were
asked to comment on the questions guiding the report for normal classroom practice (ref
Appendix XXVII). Their comments were used to review the report guidelines before use in
the pilot study. For example, comment on every task in statistics was checked, marked, had
comments and suggestions for motivation and improvement to any learning difficulty that
learners might have encountered.
57
3.5.1.7
Document analysis
In this study the documents analysed in terms of teachers’ compliance with curricular
recommendations for teaching and learning school statistics were the learners’ class
workbooks, learners’ and teachers’ portfolios, and the NCS for mathematics. The purpose of
the analysis was to triangulate the data, using the teacher interviews, questionnaires, lesson
observation and written reports on how teachers developed their PCK in statistics teaching.
At the end of the four weeks’ teaching, these documents were made available to the
researcher. The learners’ workbooks contained completed, written classwork, homework, and remedial
work. Teachers’ portfolios for example included work samples and reflective commentary by
the teachers as to what had made the lesson easy or difficult, and intervention strategies
adopted to address learners’ learning difficulties, if any (ref Appendix XXI). The NCS policy documents gave an indication of whether the teachers were adhering to
policy recommendations for teaching and learning, such as the work schedule to be used for
teaching statistics according to grade, resources, and assessment plans. It is assumed that a
teacher with adequate knowledge of the curriculum would be able to design good teaching
strategies in line with the curricular goals. In practice, this requirement meant checking for
consistency in the implementation of lesson plans according to the NCS.
3.5.1.8
Video recording
The purpose of the video recording was to record the teachers’ teaching (lessons), which
would enable the researcher to triangulate the data collected from the lesson observations.
The duration of the lessons observed ranged from 40 to 45 minutes for each of the eight
lessons. The transcribed protocols (ref Appendix V-XII) were used to gain insight into
teachers’ content knowledge and how it was used, including the instructional strategies
demonstrated in the lessons on statistical graphs.
3.6
Validation of the research instruments
Validity tells us whether an instrument measures or describes what it is supposed to measure
or describe. It means that whatever scores were obtained from the instrument should make
sense, be meaningful, and enable the researcher to draw conclusions from the sample of the
58
population under investigation (Creswell, 2008). The test validity of an instrument could
involve construct validity, content validity, and criterion validity (Creswell, 2008). In this
study, content validity was chosen to validate the test instrument (conceptual knowledge
exercise). The purpose was to determine whether the test covered the content of the domain
that it was supposed to measure. The instrument was meant to assess the subject matter
content knowledge in statistics (the domain) that the selected mathematics teachers
possessed, which, it was assumed, enabled them to develop PCK. The other instruments such
as the concept map exercise and semi-structured interview schedule were validated as follow.
3.6.1 Validity and reliability of the concept map
The purpose of the concept mapping exercise (ref Appendix XXIV) was to assess the
participating teachers' knowledge of the school statistics curriculum. In this study, although
the major instruments used for assessing teachers' school statistics content knowledge were
the statistics conceptual knowledge exercise and teacher lesson observation, the concept map
exercise was further used as an addendum to that assessment. A concept map is a viable
means of gathering information on a person’s conceptual knowledge of a topic (Novak &
Canas, 2006). The concept mapping exercise required the participating teachers first to list
the given school statistics topics according to the grades for which those topics are taught,
namely Grades 10, 11 or 12; and second to arrange them in the order in which they should be
taught in a conceptually logical and sequential fashion. The assumption was that ability to
arrange the topics for teaching in a hierarchical manner for each grade level provided an
indirect indication that the teachers had adequate knowledge of the statistics topics in the
mathematics curriculum and the conceptual relationships among them.
A given set of criteria was used by a mathematics specialist from the Department of
Education and two university lecturers in the Mathematics Education Department (ref
Appendices XXIV and XXV) to content validate the concept map exercise and
memorandum. First, the experts had to ascertain whether the concept map exercise would
allow the mathematics teachers to list the topics according to Grades 10, 11 and 12 and
arrange them in logical order, such that one topic formed the basal knowledge of the next for
each of those grades. Second, they were required to ascertain whether the memorandum
(expected answers to the concept mapping exercise) was appropriate for answering the
concept mapping exercise. The experts’ responses (mathematics specialist from the
Department of Education and two university lecturers in the Mathematics Education
59
Department) showed unanimous agreement that the concept map exercise contained adequate
information for assessing teachers' content knowledge of the statistics topics in the various
grades and the ways in which they should be taught in a logical and sequential order. In
addition, all the raters agreed that the memorandum was adequate and appropriate for
assessing the concept mapping exercise.
The reliability of the concept map was determined as follows. The concept map exercise and
memorandum were given to four school mathematics teachers that did not participate in the
study and who were physically located outside the study site to avoid contamination. There
were consistencies in the responses of the mathematics teachers with the anticipated answers
(memorandum) of the concept mapping exercise. In other words, the responses of the
respondents (mathematics teachers) were consistent with the idea of listing the statistics
topics according to grade and the way in which they should be taught in a logical hierarchical
and sequential order. The consistency in the responses of the teachers indicated that the
concept mapping exercise is reliable enough for assessing the teachers' knowledge of
statistics in the school mathematics curriculum (Bush, 2002; Barriball & White, 2006).
Where necessary, their responses were used to review the concept mapping exercise and
memorandum before they were used for the main study.
3.6.2 Validity and reliability of the interview schedule
The purpose of the semi-structured interview (ref Appendix XXVI) was to assess the
educational backgrounds that may have enabled the mathematics teachers to develop their
assumed topic-specific PCK in statistics (Jong, 2003; Jong et al., 2005; Van Driel et al.,
1998; and Rollnick et al., 2008).
The schedule was validated by a mathematics expert in the Department of Education and two
mathematics education specialists from a university, using a specific set of criteria. The raters
were requested to establish whether the interview schedule contained appropriate information
to determine teachers’ mathematics educational background for developing PCK as defined
in statistics teaching (ref Appendix IV). Their responses showed unanimous agreement that
the schedule contained the necessary information for assessing how the participating teachers
developed their topic-specific PCK (Bush, 2002; Barriball & White, 2006).
To ascertain the reliability of the interview schedule, it was used with some school
mathematics teachers who were not participating or involved in the study. The interest was in
60
determining the extent to which the schedule was likely to yield consistent responses from
them (Bush, 2002) in terms of assessing the mathematics teachers' educational background
that may have enabled them to develop their topic-specific PCK in statistics teaching (ref
Appendix XXVI). The responses of the pilot teachers were identical and consistent in terms
of the items selected for the interview schedule. The reliability of the instrument was thus
generally assured and, where necessary, the respondents’ comments were used to review the
schedule.
3.7
Pilot study
Purpose of the pilot study
The purposes of the study were:
•
To test the validity and reliability of the test instruments
•
To test the logistics feasibility for administration of the instruments
•
To improve the design of the research instruments and methodology for the
administration of the main study
•
To check that the instructions given to investigators were comprehensible
•
To check the timing for the administration of the instruments
3.7.1
Subjects used in the pilot study
The subjects used in the pilot study were two willing mathematics teachers at high school
level who did not participate in the main study. They were selected from their schools’
performance in mathematics in the Senior Certificate Examination for at least two years. The
participating teachers had taught higher grade or optional mathematics for a minimum period
of three years. One of the participants had a BSc (Hons) in mathematics and the others had
BEd degrees in mathematics education. All the participants had taught mathematics at high
school for a minimum of five years. The schools from which the participants were selected
had shown consistent pass rates of 70% and above in mathematics for at least two years.
3.7.2
Administration of the pilot study
The researcher applied for permission to administer the test to the teachers from the
Provincial Department of Education (ref Appendix III). Permission was granted and the
teachers participated voluntarily in the exercise (ref Appendix I). 61
The conceptual knowledge exercise, concept mapping, lesson observation schedule, lesson
plan schedule, questionnaire, interview schedule, teachers’ written report guide, video
recording and document analysis schedule were administered to the participants during the
pilot study. The conceptual knowledge exercise was administered to the participants in a
classroom at the centre where the cluster meeting took place. Before the conceptual
knowledge exercise was administered, participants were informed of their right to participate
voluntarily or withdraw from the research process if they wished to, and were informed of
their role, the aims and objectives of the research, and how their privacy would be
maintained. The time for the completion of the conceptual knowledge exercise ranged from
45 to 55 minutes.
3.7.3
3.7.3.1
Result of the pilot study
Conceptual knowledge exercise
As indicated in Section 3.5.1.1, three mathematics lecturers (raters) from the university
assessed the first draft of the conceptual knowledge exercise for content validity. Content
validity was obtained by determining the extent to which the raters agreed with the researcher
(test developer) and whether the test covered the entire content of statistics in school
mathematics adequately according to the NCS (ref Appendix XXIX). The raters were asked
to rate each question in terms of sureness (with rating levels of; 1 = not very sure; 2 = fairly
sure; and 3 = very sure) and relevance (with rating levels of; 1 = low/not relevant; 2 = fairly
relevant; 3 = highly relevant), with a maximum of three marks for each question. By
indicating ‘sureness’, one had no doubt that the instrument measured the content knowledge
of the chosen topic. By indicating ‘relevance’, one had no doubt that the item was a measure
or determinant of content knowledge of the chosen topic (ref Appendix XXIX). The raters’
responses demonstrated an overall average of 97% agreement (for the first draft) on the
extent to which the test items covered the curriculum. Furthermore, based on their comments,
the final items agreed upon totalled 20. Additionally, the instrument was given to some Grade 11 and 12 mathematics teachers who
would not participate in the conceptual knowledge exercise in order to identify difficult and
confusing terms or phrases and these were modified or rephrased. • Scoring the conceptual knowledge exercise 62
Marks were allocated for correct responses or correct choice of options, and no mark was
allocated for a wrong or omitted choice, or a choice of more than one response per item. The
total correct score was determined out of 20 and the percentage of the score was calculated.
Both the raw score and percentage score were analysed to determine the reliability of the
exercise. Other test characteristics, including the item response pattern, discrimination, and
difficulty indices, were determined and are discussed below.
a)
Item response pattern
The analysis of the conceptual knowledge test showed that in some of the items, only 1 or 2
or 3 or 4 (both) participants chose the items, as shown in Table 3.6. For instance, all the
participants chose option E of item 1, which is the correct answer to the item. In item 3, three
participants chose option B and only one participant chose option D. In items 12 and 13, only
one participant chose option D in each case. One of the participants wrote ‘no answer’ and
the other two left the question unanswered. This may be due to bad distracters. Such
ambiguous items were discarded. All the participants answered items 1, 2, 5, 8, 9, 19, 11, 14,
15, 16, 17, 18, 18 and 20 correctly. These items tested participants’ knowledge in statistics in
school mathematics according to the NCS. While items 1, 2, 11, 15, and 20 tested
comprehension, items 5, 8, 9, 10, 14, 16, 17 and 18 tested application. These items seemed to
be easy for the teachers. The items were modified by replacing and rephrasing the questions
and were therefore considered for inclusion in the main study. Few participants answered
items 2, 3, 4, 6, 7, and 19 correctly. These items tested teachers’ knowledge of measures of
central tendency in statistical graphs of grouped data. While items 2, 3 and 6 tested
comprehension, in which the teacher applied routine procedures to solve graphing problem in
familiar context, items 7 and 19 tested application, in which the teacher applied his
knowledge of statistics to solve familiar or other situations by using routine or multi-step
procedures. These items were considered difficult. At the end of the review exercise, based
on test characteristics (item response pattern), 20 items testing statistics in school
mathematics (measures of central tendency and spread) were selected for the main study.
63
Table 3.8:
Item response pattern of the conceptual knowledge exercise from the pilot
study test items
OPTION
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
A
0
0
0
0
0
3
3
4
0
0
4
0
0
0
4
4
0
4
1
4
B
0
0
3
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
2
0
C
0
0
1
3
4
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
D
0
4
0
0
0
0
1
0
0
0
0
1
1
4
0
0
0
0
1
0
E
4
0
0
1
0
1
0
0
4
0
0
0
0
0
0
0
0
0
0
0
b)
Reliability of the conceptual knowledge exercise
Reliability is the extent to which a test produces similar results when administered under
constant conditions on all occasions (Cohen, Manion & Morrison, 2007). It refers to the
ability of a researcher to obtain the same response each time a test is administered.
Principally, there are three types of reliability: stability, equivalence and internal consistency
reliability (Creswell, 2007). While stability and equivalence can be examined by test-retest
procedures (to give the same test to the same group on different occasions), internal
consistency can be examined using the Kuder-Richardson split half procedure (KR-20, KR21) or coefficient alpha (Creswell, 2007). Reliability in terms of stability and internal
consistency of the conceptual knowledge exercise was established in this study using the
Kuder-Richardson split half procedure (KR-20, KR-21).
In measuring the stability using the test-retest method, the scores of two tests from two
similar groups were correlated. The correlation coefficient must be significant at 95% or a
higher confidence interval (Cohen, Manion and Morrison, 2007). In this study, similar groups
of teachers were used to pilot test the instrument in order to establish the reliability of the
instruments.
The correlation (r) of the two equivalent groups was determined with Window SPSS Version
17.0 as shown in Table 3.11.
r = N ∑ XY − (∑ X )(∑ Y )
[N ∑ X
2
][
− (∑ X ) N ∑ Y 2 − (∑ Y )
2
2
]
Where:
64
r = the correlation between the two half (even numbered and odd-numbered) items
N = total number of scores
∑X = sum of scores from the first half test (even-numbered items)
∑Y = sum of scores from the second half test (odd numbered items)
∑X2 = sum of the squared scores from the first half test
∑Y2 = sum of the squared scores from the second scores from the second half test
∑XY = sum of the product of the scores from the first and second half test
Applying the Spearman Brown prophecy formula to adjust the correlation coefficient, (R)
was obtained to reflect the full-length exercise (Creswell, 2007; Gay, 1987; Gall and Borg,
1996);
R =
2r
1+ r
Where:
R = estimated reliability coefficient of the full length exercise
r = the correlation between the two half length exercises
The actual correlation (r) between the two half-length exercises was found to be 0.70. Hence,
the reliability coefficient (R) of the test is 0.81. The reliability coefficient is within the limit
of the acceptable range of reliability 0.70–1.00 (Adkins 1974; Hinkle, 1998). The exercise
that was developed can therefore be considered reliable for use in the main study. c)
Discrimination index
The discrimination index is a measure of the effectiveness of an item in discriminating
between high and low scorers on the whole test (Tristan, 1998). Once a discrimination index
of an item has been computed, the value can be interpreted as an indication of the extent to
which overall knowledge of the content area is related to the responses on an item. Therefore
it is considered that the ability of a test taker to answer an item correctly depends on the level
of knowledge that the test taker has about a subject or topic.
65
The following statistical formula was used to determine the (DI) of the conceptual knowledge exercise. D =
R − RL
RH RL
OR D = H
(if nH = nL)
−
nH nL
N
where:
D = item discrimination index
RH = number of teachers from the high scoring group who answered the item
correctly
RL = number of teachers from the low scoring group who answered the item
correctly
nH = total number of high scorers
nL = total number of low scorers
The discrimination index of each item was obtained by subtracting the proportion of low
scorers who answered the question correctly, from the proportion of high scorers who
answered the question correctly (Trochium, 2001). The discrimination index is a measure of
the quality of the items in the exercise and identifies the teachers who possess the desired
competency as well as those who do not. The discrimination index ranges from -1.0 to +1.0.
If the discrimination index is positive, it means that more test takers in the higher group
answered the item correctly than the test takers in the lower group. Table 3.9: Summary of discrimination indices of the test items
Item no
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Discr index
1.0
0.5
0.5
1.0
1.0
1.0
0.5
0.0
1.0
1.0
1.0
0.0o
0.0
1.0
1.0
1.0
1.0
1.0
0.0
1.0
In this study, a discrimination index range of 0.5 to 1.0 was considered appropriate for the
inclusion of items in the test instrument (Haladyna, Downing & Rodriguez, 2002). It was
therefore necessary to choose more difficult items since the researcher was interested in
assessing the content and competency of the teachers in the topic. All the items (e.g.
questions 8, 12, 13, and 19) outside the range 0.5 to 1.0 were modified, replaced, or
discarded. The overall discrimination index was 0.7, which was within the acceptable range
of values for the test characteristics.
66
(d) Index of difficulty
Another statistical technique that was applied to determine the quality of the test was the
index of difficulty. The index of difficulty is given by:
P=
R ∗ 100
where; P = index of difficulty
n
n = total number of teachers in the high and low
scoring groups
R = number of high and low scoring teachers who answer
the item correctly The index of difficulty was determined by calculating the proportion of the participants
taking the test who answered the item correctly (Nitko, 1996). The larger the proportion, the
more students who have learned the content measured by the item (Haladyna, et al., 2002). A
test with an overall index of difficulty of more than 0.8 is considered too easy (Nitko 1996).
In this study, a difficulty index range of 0.4 to 1.0 was considered appropriate for the
inclusion of an item in the test instrument, since participants were assumed to be competent
in this topic and were currently teaching it. It was therefore necessary to modify, replace,
simplify, or discard items that were outside this difficulty index range. Table 3.10 below
summarises the difficulty indices of the tests items.
Table 3.10:
Summary of difficulty indices of the test items
Item no
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Diff. index
1.0
0.3
0.5
0.8
1.0
0.8
0.5
0.0
1.0
1.0
1.0
0.0
0.0
1.0
1.0
1.0
1.0
1.0
0.0
1.0
The above analysis shows that about 80% of the items had difficulty indices of between 0.5
and 1.0. This is within the acceptable range. The items cover mode, median, mean, pie charts,
histograms, double bar graphs, ogives, variance, standard deviation and scatter diagrams.
Therefore, most of the items were retained for use in the main study. Items 2, 12, and 13,
with difficult indices of less than 0.4, cover some aspects of grouped data and double bar
graphs. The overall difficulty index was 0.7, which was within the acceptable range of values
for the test characteristics. Table 3.11 shows the summary of the test characteristics for the
conceptual knowledge exercise.
67
Table 3.11:
Summary of test characteristics
Range of values for test
characteristics
Results from pilot study
0.70 to 1.00
0.81
Discrimination index
0.3 to 1.0
0,7
Index of difficulties
0,4 to 1,0
0,7
0,97
0,7
Test characteristics
Reliability
Content validity
3.7.3.2
Concept mapping
With reference to Section 3.5.1.2, the concept map and method of assessing (memorandum)
the responses of the participating teachers were validated by mathematics specialists from the
Department of Basic Education and two lecturers from the university (ref Appendix XXIV)
on the content of statistics in school mathematics according to the NCS.
As explained in Section 3.5.1.2, it is assumed that if participants are able to group the
statistics topics according to how they should be taught in each grade and in a sequential
fashion, then they possess sufficient knowledge of the curriculum and how it should be
organised for effective teaching. The raters’ responses showed that, first, the list of contents
of NCS statistics adequately covered the contents of statistics in school mathematics in
accordance with the National Curriculum Statement for Mathematics (DoE, 2010). Second,
the memorandum developed for scoring was said to be adequate to assess the performance of
the participating teachers in the concept map exercise. Therefore, the concept map instrument
was accepted for the pilot study. Using the rubric designed to evaluate the concept map
drawn by the participants, the first participant scored 62% and the second participant 64% in
the pilot study (see method of scoring in Section 3.5.1.2).
3.7.3.3
Lesson observation schedule
As discussed in Section 3.5.1.4, two mathematics lecturers at the university validated the
lesson plan and lesson observation schedules adopted from the Provincial Department of
Education. Criteria (ref Appendix XVI) were developed by the researcher by taking into
consideration how a standard lesson in normal classroom practice is supposed to proceed
(Ofsted, 2010). The two lecturers were questioned, via these criteria, to determine whether
68
the lesson plan and observation schedule contained adequate information to assess normal
classroom practice in compliance with the NCS. The validation confirmed that the schedules
were the current ones used by mathematics teachers according to the NCS and contained the
necessary information to assess normal classroom practice. The classroom observation
schedule was used by the researcher during the observation of lessons (ref Appendix XVI).
The lesson observation schedule contained information such as planning, which involves the
lesson topic, learning outcomes, assessment standard and resources. The second part
described pedagogical issues, such as the introduction of the lesson, general class handling
involving class organisation, discipline, interaction, movement, learning climate and
involvement of learners in the lesson. Other pedagogical issues contained in the lesson
observation schedule were the lesson development, consolidation of the lesson and
description of actual teaching and learning. In the actual teaching and learning, the language
used for teaching, questioning techniques, assessments, use of resources, knowledge of the
teacher, and errors and misconceptions identified were included in the schedule. The teachers
and learners’ activities, as well as how the lesson was evaluated before the conclusion, were
also contained in the lesson observation schedule (ref Appendix XXXII).
3.7. 3.4
Interview schedule
The mathematics expert and the two lecturers were requested to assess the interview schedule
to determine if it contained adequate information that will enable the researcher to gather data
to gain an insight into the mathematics teachers’ content knowledge and educational
background for developing PCK in statistics teaching. Their responses to the items in the
interview questions showed that the schedule contained adequate information needed to
assess teachers’ PCK. The items that were not well phrased were modified before they were
used in the pilot study (ref Appendix XXVI).
3.7.3.5
Questionnaire for teachers
As explained in Section 3.1.5.2, the questionnaire for the teachers focused on what the
teachers did while teaching, namely the strategies used or approach/methods applied, the
content of the lessons, the nature of the topic, how the teachers identified the learners'
preconceptions and learning difficulties, how the difficulties were resolved if they were, and
how the lessons were evaluated. The two lecturers and the mathematics expert validated the
designed questionnaire by the researcher with the aid of several sources on classroom
69
practice (Leinhardt et al, 1990; Muijs & Reynolds, 2000; Cangelosi, 1996; Erickson, 1999;
DoE, 2010). The two lecturers and the specialists were requested to assess if the
questionnaire adequately covered what the teachers are supposed to do while carrying out
effective teaching in classrooms with a set of criteria (ref Appendix XIII). Their reports
showed that the questionnaire contained questions that are able to elicit from the teachers’
information regarding the instructional strategies that mathematics teachers use during
classroom practice. The comments from the mathematics specialists and English specialists
were used to review the questionnaire before it was used in the pilot study (ref Appendix
XXVIII).
3.7.3.6
Written report guide
The written report guide (ref Section 3.5.1.6) was validated by a mathematics specialist in the
Department of Basic Education and two lecturers from the university. The three of them were
requested to determine whether the written report guides could be used to collect data about
what has made the lesson easy or difficult with a set of criteria (ref Appendix XIV). Their
responses confirmed that the guide contained adequate questions to guide a mathematics
teacher to write such a report. Their comments were used to revise the written report guide
before it was considered for the pilot study (ref Appendix XXVII).
3.8
3.8.1
Main study
Subjects used in the main study
The selected four mathematics teachers at high school level in Tshwane North District were
involved in the main study. 3.8.2
Administration of the main study
The procedure used in administering the pilot study was also used for the main study. The
validated test instruments consisting of i) conceptual knowledge exercises; ii) concept
mapping; iii) interview schedule; iv) lesson plan and observation schedule; v) questionnaires;
vi) teacher written report guides, vii) and document analysis schedule were administered to
the participants. The teachers taught for four weeks and eight periods of lessons were
observed on scheduled dates by the researcher. 70
3.9
Data analysis and results of the main study
3.9.1
Quantitative data analysis
The scores obtained by the four teachers who wrote the conceptual knowledge and concept
mapping exercises in the main test were scored as described in sections 4.2 and 4.4 of this
study.
3.9.2
Qualitative data analysis
The qualitative data gathered from teachers using the teacher interview, questionnaire,
written report and document analysis were analysed by coding and categorising their
responses according to the theme in order to determine how the participating teachers
developed their PCK in statistics teaching. The analyses were described in Section 4.7.
For the lesson observation, the duration of the observed lessons ranged from 40 to 45 minutes
with each of the four participants, and the observation was conducted over four weeks of
teaching statistical graphs. The purpose of lesson observation was to determine the subject
matter content knowledge, knowledge of instructional skills and strategies as well as insight
into learners’ conceptions and learning difficulties that the teachers demonstrated in
classroom practice over the period. The lesson observations were analysed using the format
and content of the lesson observation schedule designed by the Department of Basic
Education for normal classroom practice, as was done in the pilot study. The lesson
observation reports were coded and categorised in order to determine the similarities and
differences between the teachers’ teaching methods in the assigned topic (statistical graphs). The reports of the lesson observations for each of the participants allowed for individual
lesson observation analysis and comparison of the instructional skills and strategies used for
teaching school statistics. The similarities and differences in content knowledge, knowledge
of learners’ conceptions in the learning of statistics and learning difficulties that the
participating teachers demonstrated and enabled them to develop topic-specific PCK in
statistics teaching.
71
3.10
Ethical issues
Before the commencement of data collection for this study, the researcher applied for ethical
clearance. The application was approved and the researcher was issued with a clearance letter
(ref Appendices 1, 2, 3A & 3B). The participants in this study were duly informed of the objectives of the study in writing and
oral explanation before the tests were administered to them (ref Appendix I). All the
procedures that involved the participants were explained to them. They were informed of
their right to decline participation in the study if they so wished. The schools and participants
were given codes to ensure that they remained anonymous to the public. The test scripts,
interview schedule, responses to questionnaires, the CD for the video recording, and the
written reports were kept in a safe place after the information was used for this study. The
performance of the participants in the conceptual knowledge exercise was highly
confidential. Participants and participating schools were promised access to the result on
request. The study report will be submitted to the supervisor of the study, the Gauteng
Department of Education, and the University of Pretoria. 3.11
Summary of the chapter
The piloting process of this study was conducted in two phases. The first phase consisted of
development, administration and writing of the conceptual knowledge exercise with willing
participants. The second phase was to administer and validate the research instruments. The
feedback from the pilot study showed that the conceptual knowledge exercise and concept
mapping needed to be modified before the main study was undertaken. Other instruments
such as the interview schedule, the lesson observation schedule, the lesson plan schedule, the
questionnaire and written report were found to contain adequate information that could be
used to assess subject matter content knowledge, educational background, instructional skills
and strategies as well as knowledge of learners’ learning difficulties that teachers use in
teaching statistics in school mathematics. The administration of the main study followed the
procedure used in the pilot study. 72
CHAPTER FOUR
4.0
DATA ANALYSIS AND RESULTS
4.1
Introduction
This chapter contains an analysis of the data and presents the results of the main study.
Statistical procedures (outlined in Sections 3.7.3.1 and 3.7.3.2) were used to analyse the
quantitative data by categorising the responses and lesson observations of the participating
teachers according to the components of PCK (pedagogical content knowledge) in order to
answer the research questions. The results are presented in the following order:
4.2
•
Conceptual knowledge exercise
•
Concept mapping exercise
•
Classroom practice (lesson observation)
•
Teacher interview
•
Teacher questionnaire
•
Teacher written report
•
Classroom observations and video recordings
•
Document analysis
Conceptual knowledge exercise
The main purpose of the conceptual knowledge exercise was to make a performance-based
selection of teachers for the second phase of the study. The second phase consisted of a
concept mapping exercise, an interview, lesson observations, questionnaires, written reports,
and document analyses.
The percentage scores of the top four teachers, designated A, B, C, and D, in the conceptual
knowledge exercise were 85, 90, 90, and 75 respectively.
4.3
Teacher demographic profiles
The profiles of the four selected teachers are presented below.
73
Table 4.2:
Name of
teacher
Teacher A, B, C, and D profiles
Qualification
Subject taught
Teaching experience
(in years)
Grade
taught
Teacher A
BEd (Mathematics
Education), BA
(Psychology), Diploma
(Mathematics and Science)
Mathematics
21 years
11 and 12
Teacher B
BSc (Mathematics and
Statistics)
Mathematics &
Mathematical Literacy
10 years
11 and 12
Teacher C
BSc (Mathematics)
Mathematics &
Mathematical Literacy
5 years
11 and 12
Teacher D
BEd (Mathematics
Education), SED
(Mathematics and
Biology)
Mathematics
15 years
11 and 12
It is clear that the participants are qualified and experienced mathematics teachers and it was
assumed that they have sufficient subject matter content knowledge to competently teach
statistics in school mathematics.
4.4
Concept mapping
The four teachers drew a concept map (ref Section 3.5.1.2) on statistics. The results of this
exercise, assessed according to the guidelines used to evaluate their responses (ref Section
3.5.1.2), showed that teachers A and C scored 100% each; Teacher B scored 92%; and
Teacher D scored 80%. Teachers A and C arranged their topics according to the scheme used,
so no marks were deducted. Teachers A and C had greater knowledge than teachers B and D
of the school statistics curriculum content and how it should be taught logically so that one
topic formed the basal knowledge for the next topic.
4.5
Classroom practice (lesson observation)
The purpose of the lesson observation was to examine interaction patterns in the classroom
for each of the teachers, in other words how they used their content knowledge in teaching
particular statistics topics. The instructional skills and strategies used by the teachers, the
74
ways in which they tried to identify learners’ preconceptions and learning difficulties, and
what they did to rectify these misconceptions and learning difficulties, if any, were also
examined. The topic in which most lessons were observed was graphing in statistics (line
graphs, bar graphs, histograms, pie charts, frequency polygons, ogives, box-and-whisker
plots, and scatter plots) since this topic is one of the most challenging in school statistics
(DoE, 2010). Two periods of lessons were observed at a time, during site visits to each of the
teachers. The observations focused on what the teacher did before (e.g. lesson planning),
during (e.g. asking oral probing questions to determine learners’ prior knowledge), and after
the lesson (e.g. post-teaching discussions and other interventions to address identified
learning difficulties).
The same format of analysis was used for all the teachers to identify the components of PCK
used in teaching the lessons. The next section presents an analysis of the lesson observation
of Teacher A. While observing the teachers, the focus was on how the teachers demonstrated
their content knowledge, pedagogical knowledge, knowledge of learners’ preconceptions and
learning difficulties such as indicating how their assumed PCK manifested during classroom
practice. The analysis of the lesson observation will also take into account the coding and
categorisation of the themes as shown on the table.
4.5.1
Lesson observation of Teacher A
This section briefly describes Teacher A’s lesson observations on teaching statistical graphs.
The lesson focused on the construction, analysis, and interpretation of histograms and boxand-whiskers plot respectively. The condition of the classroom is first described, followed by
the teacher’s classroom practice.
75
Table 4.5.1
Description of classroom condition and lesson observation of Teacher A
DESCRIPTION OF LESSON
CATEGORISATION/THEMES
1)
Condition of the classroom
There were 15 male and 20 female learners of mixed ability. Learners were comfortably seated in six columns of
single chairs and desks, with sufficient space to move between the desks. The teacher had a full view of the entire
class during the lessons. The classroom walls were decorated with science wall charts. The furniture, windows and
The classroom presented a safe learning
environment for both boys and girls.
2)
Learners were well resourced with textbooks and
other learning materials including workbooks.
door were in good condition, with electrical wiring that permitted the use of appliances such as an overhead
projector. The mathematics class was resourced with textbooks, calculators, exercise books, and graph sheets for
each learner, as well as construction instruments for the teacher (ruler, protractor, and pair of dividers).
The classroom had locks, and burglar bars for supervised entry
CLASSROOM PRACTICE (FIRST LESSON OBSERVATION)
CATEGORISATION/THEMES
Topic: Construction, analysis, and interpretation of a histogram. Class: Grade 11
Line 1: After Teacher A had greeted the class, he introduced the lesson on histograms with oral questioning,
Oral probing questioning was used as an instructional
distributed evenly to different learners, and requested them to define the mode, the median, and the mean in a
strategy (pedagogic knowledge) to introduce the lesson
distribution of ungrouped data
on histograms and determine learners’ conceptions and
definitions of basic concepts linked to the grouping of
data in histogram construction (line 1)
Line 2: One of the learners defined mode as: ‘..... The number that appears most often in a distribution,’ and gave
Learners correctly defined mode, median and mean
an example of mode by verbally listing some numbers and locating the mode within the listed numbers. A second
(line 2), attesting to teacher A’s content knowledge.
learner defined the median as: ‘… The middle number when a distribution of numbers is arranged according to
Using a questioning strategy, Teacher A was able to
size.’ A third learner defined the mean as: ‘… The average of the distribution.’ The last answer was followed by an
identify learners’ previous knowledge about the
example from the same learner, who listed some numbers, added them all together, and divided the sum by the
statistics lesson topic.
number of numbers on the list, to determine the mean. All three learners identified or pointed out by the teacher
provided correct definitions for the terms ‘mode’, ’median’, and ’mean’.
76
Line 3: After the introduction, Teacher A gave the class an example of how to construct and interpret a histogram.
Teacher content knowledge was used to work through
He said, ‘Write down this example.’ : (i) Construct a frequency table of five classes, starting from 16, (ii) calculate
an example of how to construct and interpret a
the mean, (iii) draw a histogram, and (iv) use the histogram to calculate the mode of the ages, correct to the nearest
histogram (line 3)
year, of 27 members of a netball club. The ages are as follows: 17, 21, 23, 19, 27, 38, 20, 21, 28, 31, 18, 21, 24, 30,
25, 19, 22, 27, 35, 18, 27, 22, 20, 30, 27, 21, and 23. The solution to these questions was presented as follows by
Teacher A and the learners, working together:
(1)
Teacher content knowledge was used to describe and
Construction of frequency table
Line 4a: Teacher A drew a frequency table with the given class intervals, as shown in table 4.5.1a. The table
contained the ages of the members of a netball club, the frequencies of the age groups, the mid-values (x) of the
complete a frequency table from raw data (lines 4a and
4b)
age groups, the class boundaries and fx. The teacher did not explain the meaning of the terms. It may be assumed
He engages learners by asking them to determine the
that the learners had come across terms such as class interval (ages), frequencies, mid-values, and the product of
frequencies within the class intervals row by row (line
frequency and mid-values before because preparing a frequency table of ungrouped data is taught before grouped
4b).
data according to the curriculum. Teacher A showed the learners how the class intervals (ages) are calculated,
using a class of five: for example, he said, ’Beginning from 16 and with five classes, the next class is 20. Therefore,
16–20 is a class interval.’ The teacher continued, ‘The next class is 21–25, the other class intervals are: 26-30, 3135, and 36-40’ (see Table 4.5.1a).
Line 4b: Teacher A listed the frequencies of the frequency (f) column on the chalkboard as learners individually
counted the ages within the intervals (see Table 4.5.1a) under his instruction. For instance, he asked, ‘How many
persons are within the ages 16-20?’ The learners counted individually and indicated the frequencies to the teacher
who wrote them in the frequency column.
77
Table 4.5.1a. A frequency table showing the age distribution of members of a netball club
Ages
Freq. (f)
Mid-values (x)
fx
Class boundaries
16–20
6
18
108
15.5–20.5
21–25
10
23
230
20.5–25.5
26–30
8
28
224
25.5–30.5
31–35
2
33
66
30.5–35.5
35–40
1
38
38
35.5–40.5
27
666
Line 5a: Teacher A showed the learners how to calculate the mid-values: e.g. he said, ‘Mid-value =
(for the first row).’ Teacher A continued, ‘For the second row: mid-value =
16 + 20
= 18
2
21 + 25
= 23 (for the second row)
2
.Now continues with row 3, 4 and 5.’ The learners continued with the calculation of mid-values while the teacher
wrote the acceptable values on the chalk board.
Line 5b: The next step was to calculate fx, meaning frequency multiplied by mid-values (x). Teacher A
demonstrated: ‘To calculate fx, you multiply the value of frequency and mid-values, i.e. fx = 6 x 18 = 108 for the
first row; for the second row, fx= 10 x 23 = 230; for the third row, fx = 8 x 28 = 224; for the fourth row, fx = 2 x
33=66; and the fifth row, fx = 1x38= 38.’
78
Teacher content knowledge was used to describe how
to calculate mid-values and fx (lines 5a and 5b).
Learner content knowledge was used to complete
mid-values (line 5b).
Line 6: Teacher A began by describing how to find the class boundaries, beginning with the first row. He then
selected an example from table 4.5.1 and calculated the lower class boundary =
15 + 16
= 15,5 (for the first
2
row). In the learners’ mother tongue, he said, ‘15 tlhakanya le 16 arola ka 2, e lekana le 15.5; Meaning add 15 to
16 and divide by 2, equal to 15.5.’ He continued: ’The upper class boundary =
20 + 21
= 20,5.’ (see table
2
Teacher content knowledge was used to describe how
to complete the frequency table by calculating midvalues and class boundaries to construct the histogram
(lines 5a and 5b). The learners’ mother tongue
(instructional strategy) was used to further reinforce a
point on how to calculate class boundaries (line 6).
4.5.1a.). Teacher A Further requested the learners to complete the class boundaries for other rows.
Line 7: The learners completed the table after the teacher had shown them how to calculate the frequencies, mid-
Teacher content knowledge was used to demonstrate
values (x), fx, and class boundaries.
how to complete the frequency table by calculating the
frequencies, mid-values, class boundaries, and fx (Line
7).
Teacher A indicated that the next exercise would comprise
Teacher content knowledge was used to calculate
mean from the frequency table (line 8) using a
(II) Calculating the mean from frequency table:
procedural knowledge approach. Procedural
to begin the demonstration on how to calculate mean from the frequency table.
knowledge approach is the skill in carrying out
Line 8: Teacher A wrote on the chalkboard: ‘Mean is calculated by using the formula,
∑ fx
∑ f
procedures flexibly, accurately, efficiently, and
, where ∑fx means
appropriately to accomplish a given mathematical task.
It includes, but is not limited to, algorithms (the
the sum of frequencies(f) multiplied by the mid-values (x) and ∑f, means summation of frequencies only, as shown
step-by-step routines needed to perform arithmetic
in Table 4.51a.’Using the formula, he showed the learners how to calculate the mean as follows:
operations).
∑ fx = 666
Mean =
∑ f 27
An algorithmic approach was used to calculate the
= 24,67
mean from the frequency table (line 8).
79
(iii)
Constructing the histogram
Teacher A defined a histogram and described how to
Line 9: Teacher A defined a histogram as: ’… a statistical graph which is used to represent grouped data; a
histogram helps to understand complex data in a simpler manner through visualisation.’ He then described how to
construct a histogram without explaining what the term grouped data meant. He began by drawing the horizontal
construct a histogram using a procedural as opposed to
conceptual knowledge approach. A Procedural
knowledge is a formal symbolic representation system
and vertical axes on the chalkboard and reinforced the terms using the learners’ mother tongue. He said, ‘Thala
of a given mathematical task using algorithms, or
mola o o horizontal le o o vertical’, meaning, draw the horizontal and vertical axis. This was followed by stating
rules, to complete the mathematical tasks (Star,
the chosen scale. He indicated that the scale was chosen by considering the highest and lowest values of the
2002). As indicated above, the participating teachers
frequencies and data values as well as the dimension of the graph paper provided, but without demonstrating it
used more of a procedural knowledge approach than a
mathematically to the learners. He continued with the labelling of the axes and said: ‘O be o tsenya di nomore mo
conceptual knowledge approach because the topic
meleng’, meaning, label the axes. He drew the first two bars of the histogram. He instructed the class to complete
required a particular procedure. It is the common way
the graph and stated the chosen scale again with no mathematical explanation of how the scale was chosen. To do
in which the teachers used algorithms or rules to
so would have required a more detailed conceptual explanation.
complete statistics task.He did not explain what was
meant by grouped data. Once again the mother tongue
equivalent of the technical terms was used to enhance
comprehension. Topic specific graph construction skills
of drawing horizontal and vertical axes, choosing a
scale’ and labelling the axes were used to teach the the
learners histogram construction (line 9). Teacher A
stated and used a chosen scale for constructing the
histogram without a conceptual explanation of how it
was done (line 9).
Line 10: The learners completed the histogram individually in their workbooks after the teacher had demonstrated
Learners completed the histogram based on the
on the chalkboard (ref Figure 4.5.1a) with the assistance of another learner how to construct a histogram from the
teacher’s demonstration of histogram construction on
grouped data given.
the chalk board (line10)
Line 11: Some learners seemed to have understood how to construct a histogram for they completed the exercise in
Some learners experienced difficulty in selecting the
80
their workbooks correctly. Others had difficulties in choosing an appropriate scale so that the histogram could not
appropriate scale (line 11) for constructing a histogram.
be accommodated on the graph paper provided. The teacher identified those who were experiencing difficulties
Insufficient explanation was provided by Teacher A
because these learners were erasing and correcting their mistakes. He intervened by asking one of the learners,
about how to choose scale for constructing a histogram
‘Why are you erasing your work?’ The learner answered ’My work is not correct compared to the one on the
(line 11). Learners who were experiencing some
chalkboard,’ Teacher A then asked, ‘Do you understand why your diagram is wrong?’ The learner answered ‘Yes,
difficulties
I have seen it on the chalkboard’ and the teacher directed the same question to the other learners who were also
constructed by the teacher and the learners on the
erasing their work. They all agreed that they had detected their mistakes from the correction on the chalkboard.
chalkboard (line 11).
corrected
them
with
the
histogram
The teacher had to allow the learners to write the corrections from the chalkboard into their exercise books for a
few minutes before proceeding to calculate the mode from the histogram. The intention of allowing the learners to
complete the diagram was to ensure that all participated in using the same diagram to calculate the mode. The next
histogram (line 11).
part of the lesson was on how to calculate mode from the histogram.
Figure 4.5.1a:
The learners grasped the rule for the construction of a
Histogram of the age distribution of members of a netball club (with a continuation line
from the vertical axis)
Line 12a: After the histogram was constructed by Teacher A and the learners, the teacher described another
Teacher content knowledge was used to explain
method of constructing histograms. This method allows the histogram to be constructed without a continuation line
another method of constructing a histogram. It involved
from origin of the data axis even if the data does not start from o, to reinforce the learners’ understanding of
creating a continuation line beginning from the vertical
histogram construction (ref Figure 4.5.1c). He used the same rule-oriented procedural approach.
axis. The second method helped to reinforce learners’
81
knowledge of histogram construction and interpretation
(line 12a).
Teacher content knowledge was used to describe the
procedure
(procedural
knowledge
approach)
of
constructing a histogram (line 12b).
Figure 4.5.1b:
A Histogram showing the age distribution of members of a netball club with labelling of the
data axis without a continuation line starting from the vertical axis
Line 12b: Teacher A demonstrated the construction of a histogram by beginning the labelling of the data values
from the vertical axis, plotting the points, and joining the line of best fit using the same table of values and
histogram that had just been constructed. Having constructed the histogram the, next step was to show how to
calculate the mode from it..
(iv)
Calculating the mode from the histogram
Teacher A used a procedural knowledge approach to
Line 13: Teacher A demonstrated how to calculate the mode (using Figure 4.5.1a as presented on the chalkboard).
determine the mode of a histogram (line 13) without
He first drew a diagonal line from the top right-hand corner of the highest bar of the histogram to the top right-
explaining the conceptual reasoning behind the drawing
hand corner of the bar to the left of it. The next step was to draw another diagonal from the top left-hand corner of
of the diagonal lines. Conceptual understanding
the highest bar to the top left-hand corner of the next bar to the right of it. He then drew a line from the meeting
consists of those relationships constructed internally
point of the two diagonal lines to the horizontal axis and read out the mode at that point (ref Figure 4.5.1a). No
and connected to already existing ideas. It involves the
explanation was given as to how the drawing of diagonal lines leads to the determination of the mode.
understanding of mathematical ideas and procedures
and includes the knowledge of basic arithmetic facts.
82
Students use conceptual understanding of mathematics
when they identify and apply principles, know and
apply facts and definitions, and compare and contrast
related concepts. It is called a conceptual knowledge
approach when applied in teaching.
Line 14: A learner asked, ‘Why do you have to draw a diagonal? Why don’t you simply add 20 and 25 and divide
Some learners wondered why they should draw
by 2 to get the mode?’ This question was posed by the learners because some of them had done it in that way.
diagonals to locate the mode because they calculated
Many learners nodded their heads in agreement with the question.
the average of the interval of the highest bar instead of
locating the mode within the interval of the highest bar
identified (line 14). They might have experienced this
difficulty of understanding why diagonals should be
drawn before locating the mode because the teacher had
not explained the term grouped data from the
beginning.
Line 15: Teacher A tried unsuccessfully to explain why diagonals should be drawn from both bars on either side of
The teacher used procedural knowledge to answer the
the tallest bar in the histogram to calculate the mode. He said, ‘Drawing the diagonals is a procedure for
learner’s question, but the question demanded a
calculating the mode of the grouped data, and the diagonals help to locate the mode within the intervals.’ A
conceptual knowledge (explaining the relationship and
conceptual knowledge approach of explaining the relationships among the concepts in histogram construction such
mathematical connections among the concepts in
as the class boundaries, class intervals, frequency and drawing the line of best fit of a histogram should have been
histogram construction explanation), which the teacher
used to answer the question, so as to provide clarity and the answer to the question the learners asked.
did not provide at this stage (line 15).
Line 16: Teacher A continued with the learner’s question on why the diagonal should be drawn and the average of
A conceptual knowledge approach was used to explain
20 and 25 cannot be used to calculate the mode from the histogram (line 14) when he answered, ‘You cannot find
why it is not correct to add 20 and 25 in order to
the average of 20 and 25 to give you the mode, because the intervals do not contain only the numbers 20 and
determine mode. Comparing the answers obtained from
25’There are other numbers within the intervals. ‘He referred them to stem-and-leaf diagrams (drawn previously)
a stem-and-leave with the histogram (line 16) showed
83
to show how the mode was located and said ‘Open to the stem-and-leaf you drew last time and let somebody tell us
that the teacher possesses the content knowledge
how we can locate the mode.’ One of the learners raised his hand and explained, ‘23 is the most occurring number
required to teach histogram construction.
in the stem-and-leaf diagram and that is the mode.’’ Now compare the answer we got from the stem-and-leaf and
the one from the histogram, are they the same?’ the teacher asked to elicit an answer as to whether they could link
the relationship between the two methods for calculating the mode in grouped data. The learners answered in a
chorus, ’Yees sir.’
Line 17: Teacher A continues ‘Now that you have understood the procedure I have described, write it down in
Teacher A instructed learners to copy the procedure for
your notebook’.
calculating the mode on the chalkboard (line 17).
Line 18: The learners wrote the procedure for calculating a mode from a histogram in their exercise books, as
Teacher A related the application of histograms to
provided by Teacher A (see Figure 4.5.1a and 4.5.1b) and shown on the chalkboard. Using photocopied materials,
everyday life familiar situation (line 18) (instructional
Teacher A provided examples of the useful application of histograms to everyday life situations. For example,
strategy).
‘They can be used to represent the age distribution of teachers in the school and the performance of groups or
cohorts of learners in an examination’ he said.
Learners copy the procedure as written on the
chalkboard (line 18).
Line 19a: As the lesson progressedTeacher A asked one of the learners, ‘What is the difference between a
A higher level of questioning (explanation, not recall)
histogram and a bar graph?’
was used as an instructional strategy to assess how
Line 19b: A learner answered, ‘There are constant spaces between the bars in the bar graph, but there is no space
well learners had understood the lesson (line 19a).
in the histogram between the bars. Second, the bar graph is used to represent simple data and histogram is used to
Learners showed evidence of comprehension in the
represent large groups of data. Because the data that histogram represent are large, they are grouped as class
answer provided about the differences between a bar
intervals or boundaries in the frequency table. Bar graph do not contain class interval or boundaries’ This answer
graph and a histogram (line 19b).
was satisfactory to the teacher, who asked a second question.
Line 20a: Teacher A asked: ‘How can you calculate the percentage of players within the age group of 26–40 in the
Oral questioning based on application of knowledge
histogram?’ (ref Figure 4.5.1a). A few learners indicated an interest in answering the question; one was asked to
was used to assess learners’ content knowledge about
give an answer and she said. ‘You add 7 + 2 + 1 = 10 ‘(from the frequency table), ‘then divide 10 by 27 and
histogram construction (line 20a).
84
multiply by 100; i.e. the percentage of players between 26 and 40 =
10
X 100 = 37%. Therefore, 37% of the
27
Learner content knowledge: an algorithmic approach
was used to answer the teacher’s oral question on how
to calculate the percentage of players within an age
players are between the ages of 26 and 40.’
group (line 20a).
Line 21: Teacher A assigned classwork in which the learners were asked a similar question on histogram
Classwork was used to spontaneously assess how well
construction and interpretation to the one they had already done. The classwork required learners to construct a
learners had grasped the content of the lesson
histogram and use it to determine the mode and the percentage of learners who had completed a test. Table 4.5.1b
(instructional strategy to provide immediate
(below) shows the mark distribution of the test. The teacher walked around the class to monitor the learners.
feedback) (line 20b).
Table 4.5.1b:
Teacher A monitored and analysed learners’ responses
Frequency table showing learners’ performance in a test
to classwork on construction and interpretation of
a)
Class interval (%)
Frequency
histograms (line 21) to ascertain how well the learners
40–49
2
were responding to the classwork and to detect learning
50–59
6
60–69
12
70–79
8
80–89
4
difficulties and misconceptions, if any.
Draw a histogram to illustrate learners’ performance in the test.
b) From your diagram, calculate the mode.
c)
If the pass mark is 60%, calculate the percentage of learners who failed the test.
85
Line 22a: While most learners completed the class work efficiently, some could not finish it in class. The
Learning difficulties experienced by learners were
difficulties experienced were in (i) labelling of the axes with the types of grouped data provided (which began at 40
labelling and scaling of data axes of grouped data (ref
marks and not from 0 as was the case in the example which the teacher worked on), and ii) the construction
Figure
(scaling and labelling of the axes) of the histogram. Figure 4.5.1c (below) shows an example of a graph drawn by a
comprehension was evident in a learner’s statement-
learner who experienced difficulties in histogram construction. The histogram could not be accommodated on the
(line 22b).
4.5.1c)
(lines 22a and
22b).
Lack
of
graph paper provided due to incorrect scaling.
Figure 4.5.1c:
An example of an incomplete classwork exercise on histogram construction
In this graph, the scale chosen by the learner(s) could not accommodate the histogram on the graph paper; hence
part of the histogram was not represented. This made it difficult to calculate the mode and determine the
percentage of learners who failed the test.
Line 22b: A learner said, ‘My graph is not like the one you constructed on the chalkboard.’
Line 23a: Teacher A analysed what the learner had drawn and said, ‘You constructed a bar graph instead of a
A learning difficulty of constructing a bar graph instead
histogram. It is a wrong histogram.’ He continued, ‘I shall organise an extra lesson to rectify the difficulties you
of histogram was detected by Teacher A from the
are having and explain why your answer is wrong after the lesson.’ (The lesson period had expired.) After the
classwork that the learners were doing (line 23a). A
86
lesson, some learners asked the teacher to explain aspects of the lesson where they lacked clarity (post-teaching
post-teaching discussion took place after the lesson to
discussion).
help them (line 23a).
Line 23b: Teacher A gave the learners homework on construction and interpretation of histograms using their
Learners’ learning difficulties were discovered through
textbook in mathematics, to be submitted the following day. The entire lesson was based on the learners’
an analysis of classwork (instructional strategy) (line
mathematics textbooks, photocopies of mathematics-related materials, and study guides.
21 and 23a).
Homework was used as an opportunity for learners to
demonstrate their understanding of histogram
construction, and later to assess how well the learners
had understood the lesson (instructional strategies for
teaching) (line 23b).
CLASSROOM PRACTICE (SECOND LESSON)
Topic: Construction, analysis, and interpretation of ogives and box-and-whisker plots. Class: Grade 11
DESCRIPTION
CATEGORISATION/THEMES
Line 1: Teacher A began the lesson on box-and-whisker plots by checking and marking the homework on cumulative
The checking and marking of homework was used to
frequency tables and ogives (a distribution curve in which the frequencies are cumulative).
try to determine learners’ conceptions
(preconceptions) (line 1) (instructional strategy) in
box-and-whisker plot construction.
Line 2: Teacher A and the learners provided the correct answers to the homework on the construction, analysis, and
The teacher and learners together consolidated the
interpretation of a cumulative frequency table and ogive by calculating the cumulative frequencies and further
concept previously taught by providing corrections to
explaining how it was used to construct an ogive.
the difficulties the latter must have experienced while
doing the homework (instructional strategy) (line 2)
on ogive construction.
87
Line 3a: Teacher A wrote the topic (box-and whisker plots) on the board and referred the learners to photocopied
Instructional strategies such as group work were
material on ogives, from which they could interpret an ogive using quartiles obtained from an ogive. They were to
used to interpret ogives and to demonstrate learners’
work in groups of 4 to 5 learners and calculate the quartiles as a way of demonstrating their knowledge of how to
content knowledge and understanding of how to
construct an ogive.
construct an ogive (line 3a).
Teacher content knowledge and instructional
Line 3b: Teacher A said ‘Look at the photocopied paper I have given you, question 2.’ He continued and read, ’Find
strategies were used to design the task to be used to
out the percentage of learners who obtained (i) less than the lower quartile; (ii) less than the median; and (iii) less
demonstrate box-and-whisker plot construction (line
than the upper quartile; and (iv) Minimum and maximum values of the ogive.’
3b).
Calculation of quartiles from an ogive after the learners had completed the exercise
An algorithmic approach (procedural knowledge)
Line 4: The learners interpreted the ogive (it was assumed that Teacher A had provided a description of ogive
construction in the previous lesson) as the question on the photocopy indicated, with the first quartile (see definition
below)(Q1) = 20, using the formula; Q1 =
( n + 1)th to calculate the position of Q The next step was to calculate the
1.
4
second quartile (Q2) = 23, using the formula, Q2 =
the formula,
( n + 1)th
to calculate Q3; and the third quartile (Q3) = 27, using
2
3( n + 1)th
(where a quartile is a division of the data distribution into four equal parts).
4
was used by the learners to determine the quartiles
(line 4).
Teacher content knowledge was used to provide the
definition of a box-and-whisker plot with no further
explanation regarding the basic knowledge or skills to
required for the construction of the graph. The teacher
did not indicate or anticipate any possible difficulties
or misconceptions that the learners might possibly
encounter (line 4).
‘The minimum is 15 and maximum is 38 (read from the ogive,’ one of the learners said in response to the questions
on the photocopied question. Teacher A accepted the answers provided by the learners for Q1, Q2, Q3, minimum and
maximum values as correct, and said, ’Now, we are going to use these values to construct a box-and-whisker plot.’
He defined a box-and-whisker plot as ’… a graph showing the distribution of a set of data along a number line.’
With no further explanation, he went on to describe how to construct a box-and-whisker plot.
Teacher A used a procedural knowledge approach to
Construction of box-and-whisker plot
determine the quartiles which were used to construct
88
Line 5a: Because Teacher A was satisfied with the learners’ answers on the quartiles derived from the ogive in line
a box-and-whisker diagram (line 5a) (instructional
4, he used the quartile values to show the learners how to construct and interpret box-and-whisker plots. He did this
strategy).
by first drawing a number line with a scale of 1 cm = 5 units (see below). The box was drawn above the number line
using the values for Q1, (23) Q2 and Q3 (27) (Fig 4.5.1d). The whisker was then represented by a line, according to
the maximum (38) and the minimum value (15) as obtained from the ogive as shown below (Fig 4.5.1d).
Insufficient
teacher
content
knowledge
and
explanation of box-and-whisker plot resulted in
learners’ learning difficulties (line 5b).
Construction skills were used to construct a box-andwhisker plot with the quartiles obtained from the
ogive using a procedural knowledge approach (line
5a) without explanation.
Figure 4.5.1d
represents a box-and-whisker plot constructed with the values of the quartiles obtained from
the ogive
Line 5b: Some learners experienced difficulties making sense of why the minimum and maximum values of Q1, Q2
and Q3, had to be used for constructing a box-and-whisker plot. This was largely because the teacher did not explain
the meaning of this term.
Line 6: Most of the learners requested clarity on interpreting ogives. For example: how the values of the quartiles
Learners experienced difficulties as to how the
were obtained and used to construct the box-and-whisker plot. ‘Listen learners,’ the teacher said, ’it appears that
teacher obtained the quartiles. They had no
some of you do not understand the description I have given about the construction of box-and-whisker plot. Now let
understanding of how the values of the quartiles from
me give you another example from the textbook.’
an ogive were obtained and used to construct a boxand-whisker plot (line 6). The teacher resorted to the
use of the textbook to work through a textbook ogive
example.
Line 7a: Teacher A referred the learners to their textbook, unit 8 (containing examples of what they had done). Using
Teacher subject matter content knowledge was
these textbook examples (while individual learners took note of the example from their textbook), he then tried to
supplemented by the use of textbooks to provide
89
explain the mathematical connection between ogives and box-and-whisker plots. He described an ogive as ’… a
definitions on the concepts of ogive such as the
cumulative line graph and it is best used when you want to display information involving grouped data,’ He
quartiles. There was no attempt to relate the concepts
continued, ‘To interpret an ogive, quartiles are usually used. The quartile values are used to construct the box-and
being studied to any context or examples familiar to
whisker plot to provide more clarity about what the data tend to convey.’ ‘, Teacher A said. He continued ‘’Now I
the learners. When and how are ogives used for
example? The teacher does not demonstrate any
flexibility
(or
insufficient
flexibility)
in
the
approaches or methods used to present the topic (line
7a).
An example from the learners’ mathematics textbook
was used (as an instructional strategy) to provide
some clarity on how quartiles were obtained from the
ogive and used in constructing a box-and-whisker
plot (line 7a)
Teacher content knowledge (Figure 4.5.1e) was
want you to study this example in unit 8 in your textbook for five minutes.’
used to describe the interrelationship between ogives
and box-and whisker plots by reading out the quartile
Figure 4.5.1e:
Ogive showing the mark distribution of learners in an English examination
This diagram of an ogive from the learners’ textbook was used to provide another example of the way in which to
values from the ogive in Figure 4.5.1e and used to
construct a box-and-whisker plot (line 7c).
construct ogives and box-and-whisker plots. The box-and-whisker diagram (ref Fig 4.5.1e) was constructed from
information derived from the analysis and interpretation of the ogive.
Class work was used as an instructional strategy to
reinforce learners’ grasp of how to calculate quartiles
from the ogive (line 7c).
90
Line 7b: Learners studied the example for about five minutes and compared it with their previous homework box-
Learners experienced some difficulties in locating the
and-whisker plot construction to try and comprehend how the values for constructing the box-and-whisker plot had
quartiles from the data axis due to insufficient
been obtained.
learner content knowledge about scale and labelling
Line 7c: Teacher A described:’The first quartile is obtained by first locating the quartile position on the frequency
of the data axis (line 7d).
axis, draw a line from there to join the curve, and join the line to the horizontal axis to locate first quartile (Q1)’
using Figure 4.5.1e. The same procedure is used for Q2 and Q3. Teacher A asked, ‘Do you understand?’The learners
The teacher intervened regarding the errors that the
answered, ‘Yes sir.’ .As a follow up the teacher gave them a task: Using the same Figure 4.5.1d, the learners were
learners were making on their classwork and had to
asked to (i) find the estimate of a) the lower quartile (Q1); b) the median (Q2); c) the upper quartile (Q3). (ii) Find out
work with them using Figure 4.5.1e to clarify the
what percentage of the learners had obtained marks that were a) less than the lower quartile, b) less than the median,
learning difficulties.
and c) less than the upper quartile. The intent was to find out if the learners had understood how to obtain the
quartiles from the ogive, which could then be used to construct the box-and-whisker plot.
Line 7d: As the teacher monitored and analysed learners’ classwork assignment, he discovered that the majority of
the learners were unable to locate the position of the quartile from the ogive even after applying the correct formula..,
This was either because the learners lacked the knowledge and skills of scaling and labelling of data axis, or that the
teacher’s oral explanation was not sufficient for them to grasp the concept.. Teacher A said ’I can see that some of
you cannot locate the quartiles even after you have calculated the position of the quartiles. Now, let me do it with
you.’
Finding the quartiles (Q1, Q2 and Q3)
Line 8a: Using the formula for calculating the position of quartiles as in line 4 and Figure 4.5.1e, Teacher A showed
Teacher
the learners how to calculate quartile positions by the use of a ruler to trace the quartiles beginning from the
demonstrate the procedure for calculating the
cummulative frequency axis to the curve and down to the data axis to obtain : a) the lower quartile (Q1) which was
quartiles from the data axis (line 8a) in order to
52. He continued in a similar manner to obtain: b) the median (Q2) which was 63, and c) the upper quartile (Q3)
clarify learning difficulties about box-and-whisker
91
content
knowledge
was
used
to
which was 73.
plot construction.
Calculating the percentage of learners that score marks less than the quartiles
Line 8b: Teacher A said, ’ Let us solve the remaining questions,’ and continued,, ‘you calculate 25% of 120 as:
The learners’ oral questions indicated that they had
some learning difficulties concerning the formula for
25 120
x
= 30. With your ruler at 30 on the cummulative frequency axis, trace it to join the curve and down to the
calculating the median of grouped and ungrouped
100 1
data axis. Therefore, a) 25% of the learners obtained marks of less than 52%. In a similar manner, b) 50% of the
data (line 8ci) which may have been due to
learners obtained marks of less than 63%, and c) 75% of the learners obtained marks of less than 73%.’ The
insufficient teacher explanation of the differences
answers to the two questions (i) and (ii) are the same but the question was asked in two different ways.The teacher
between the way the median in ungrouped and group
probably wanted to demonstrate varieties of ways of asking questions about quartiles and provide various strategies
data is calculated (line 8cii).
of answering the question, which illustrates the teachers’ PCK.
A conceptual knowledge approach was used to
Line 8ci: A learner raised her hand and asked, ’Why is the method of calculating the median in the ogive different
address the learners’ lack of understanding of the
from the one we did last week?’ The learner referred Teacher A to her exercise book and showed him that the
differences between how to calculate the median of
method was different from what she had in her book. Some learners nodded their heads in support of the question.
group and ungrouped data (line 8cii) by comparing
But one of learners raised his hand up and he was recognised by by the teacher to answer the question: And he said
the differences between the way the median is
'In the previous example, we calculated the median of ungrouped data. But in this case we are calculating the
calculated from grouped data in the current lesson and
median of grouped data'. The methods were different, but the learners had misunderstood the ways the median is
ungrouped data from previous lesson.
calculated in ungrouped data and in grouped data. This learning difficulty may have arisen because the teacher did
not explain the difference between determining the median of ungrouped and grouped data in line 8b and in any
previous ungrouped data lesson.
Line 8cii: Teacher A explained, pointing at the previous example in one of the learners’ exercise books and directing
the whole class to the same example in their individual exercise books, ‘The previous example used ungrouped data,
92
in which you arrange the data according to size of the numbers, but the current example used grouped data in which
some data were grouped together. You cannot arrange them in the same way like the ungrouped data because, the
particular number within the groups are not known. Hence, the formula method was appropriate to calculate the
median within the class intervals or group.’
Line 9: Teacher A asked, as a way of concluding the lesson, ‘How do you calculate the first, second and upper
Oral questioning was used to assess the learners and
quartiles of an ogive? How can you use the quartiles to construct a box-and-whisker plot?’
evaluate the lesson by requesting the learners to
explain how quartiles are calculated (line 9)
(instructional strategy).
Line 10a: Several learners volunteered to answer the question; the teacher selected one who said: ‘Using the formula
Learner content knowledge mostly of a procedural
n +1
n +1
th , you can calculate Q1 position and locate Q1. Using the formula
th , you can calculate Q2 position
4
2
or algorithmic nature was used to answer the question
and locate Q2. Using the formula
3( n + 1)
th , you can calculate Q3.and locate Q3.’ (rote learning regarding the use
4
of an algorithm).
on the application of a formula (line 10a).
The learners continued with their responses to the
teacher’s question to indicate that they had grasped
the lesson (line 10c)
Line 10 b: Teacher A called on another learner to demonstrate how the values of Q1, Q2 and Q3 could be used to
construct a box-and-whisker plot.
Line 10c: The learner used the teacher’s example to answer the question in a procedural manner by
indicating:’Using the formula (pointing on the chalkboard), you calculate Q1 position by substituting the value of n.
After that the quartile position is located on the frequency axis and by drawing a line from that position to the curve
and down to the horizontal axis, you locate the first quartile (Q1).’ Q2 and Q3 were calculated in the same way, the
learner said.
Line 11a: The learners were then referred to their textbooks for homework. This required the learners to calculate the
Homework was used as instructional strategy to
quartiles from a constructed ogive and use the quartiles to construct a box-and-whisker plot. The assessment task
assess and provide feedback on learners’ conceptual
tested learners’ conceptual understanding of how to construct, analyse, interpret and apply the knowledge of box-
understanding of the lesson on box-and-whisker plots
93
and-whisker plots to a familiar situation. The homework showed that the teacher complied with the assessment
(line 11a).
guidelines and learning outcomes of data handling-but provided no examples in his teaching of the application of
those plots in contexts familiar to the learners. Obviously, Teacher A has displayed inadequate PCK in teaching boxA post-teaching discussion was used to address
and whisker plot construction at this stage..
learners’ questions and to clarify the method of
representing fractions on the box-and-whisker plot
(lines 11b).
Line 11b: A post-teaching discussion took place after the lesson. Some learners asked: ‘How do you represent the
fractions we got from the graphs during the interpretation of the ogive?’ (following the results from their
calculations). The teacher replied: ‘The fractions can be represented by rounding off to the nearest whole number.’
Line 12: The teacher promised to organise extra tutoring after school for the learners who were experiencing
A post-teaching discussion was used to address
difficulties with the construction and interpretation of ogives and box-and-whisker plots as he could not attend to
aspects of the topic which the learners did not grasp
everybody in the post–teaching discussion.
(confusion over the use of quartile values to construct
box-and-whisker plots), and additional tutoring was
proposed (lines 12).
94
Summary of lesson observation of Teacher A
Teacher A demonstrated that he has the required content knowledge to teach statistical graphs
such as histograms, ogives and box-and-whisker plots. He described, and demonstrated how
to construct, a histogram and tried to elucidate the differences between ogives and box-andwhisker plots, using a mostly rule-oriented procedural approach; but less of a conceptual
knowledge explanation. Using his procedural knowledge he followed a stepwise sequential
approach to demonstrate the construction of a histogram and box-and-whisker plot: namely
drawing of the axes, choosing a scale, labelling the axes, plotting the points, and then
drawing the line of best fit. With regard to section 8cii of the second lesson observation,
Teacher A also applied a conceptual approach in clarifying learners’ misunderstanding of
how to construct a box-and-whisker plot using the quartiles calculated from the ogive. The
conceptual approach entails explaining in detail the relationship between the quartiles
obtained from the ogive (e.g.Q1, median, and Q3) of a box-and-whisker plot (ref Section
4.5.1, second lesson observation, and line 8cii), the mathematical connections between
quartile positions and the quartiles obtained from the ogive. Teacher A used topic-specific
construction skills (as earlier defined) in statistics to construct histograms and box-andwhisker plots. Instructional skills of oral questioning, checking and marking of learners’
classroom and homework assignments were also used to try to identify learners’
preconceptions and learning difficulties in constructing histograms and box-and-whisker
plots. But the teacher identified learners’ previous knowledge of histogram and box-andwhisker plot construction using the strategy oral questioning and checking and marking of
learners’ homework. Other instructional strategies which Teacher A applied in his teaching
were the use of examples drawn from everyday familiar situations for the histogram, but for
the ogive and box-and-whisker plot he applied the mother tongue to reinforce learners’
comprehension. There was no evidence that he anticipated the difficulties learners were likely
to have in first coming across the topics of histograms and box-and-whisker plots that he
taught. For example, when he tried to identify learners’ preconceptions using oral probing
questioning on measures of central tendency, learners displayed evidence of having a
previous knowledge of histogram construction and no preconception was identified, meaning
the teacher may well not likely have knowledge of learners’ preconceptions, which would
have allowed him to address any anticipated learning difficulty.
95
From the observed lessons, it can be construed that the PCK of Teacher A consists largely of
the procedural use of rules to construct histograms and box-and-whisker plots (statistical
graphs) and, less frequently, of conceptual knowledge.
4.5.2
Lesson observation of Teacher B
This section briefly describes Teacher B’s lessons on teaching statistical graphs. The lessons,
which were observed during two periods of site visits, focused on the construction, analysis,
and interpretation of the bar graph and the ogive. The condition of the classroom is described
first, followed by the teacher’s classroom practice in delivering the lesson.
96
Table 4.5.2a:
Description of lesson observation and classroom conditions at School B
DESCRIPTION OF LESSONS
CATEGORISATION/THEMES
1)
Condition of the classroom
surrounding some big desks in two columns.
There were 16 male and 24 female learners of mixed ability. Learners were comfortably seated in the science
laboratory in two columns of single chairs surrounding some big desks with sufficient space to move between the
desks. The laboratory was safe and conducive to teaching and learning. The wall of the laboratory was decorated
Forty learners were seated in single chairs
2)
The school was safe and well protected.
3)
The science laboratory is not used exclusively
with science charts such as the human circulatory system. The learners were individually resourced with learning
for science subjects.
material such as the mathematics textbooks, exercise books and calculators. The science laboratory is sometimes
4)
used when the teacher want to use an overhead projector for demonstration.
The learners were resourced with learning
materials
CLASSROOM PRACTICE (FIRST LESSON OBSERVATION).
CATEGORISATION/THEMES
Topic: construction and interpretation of bar graphs. Class: Grade 11
Line 1: The teacher arrived in the class and greeted the learners ‘Good afternoon learners?’ Learners answered
Teacher B greeted the class and placed a pre-activity
‘Good afternoon sir’ A frequency table was used to introduce Teacher B’s first observed lesson. Learners were
on the chalk board to gain information about learners’
expected to prepare a frequency table of the scores of learners in a test. The data presented to the learners by
conceptions (preconceptions) of the construction and
Teacher B was based on the scores that learners had obtained in a 10-mark test, and involved arranging these scores
interpretation of bar graphs (line 1) (instructional
on a frequency table: 2, 3, 4, 5, 5, 6, 4, 7, 5, 6.
strategy).
Learners showed that they had assimilated the
Line 2: The learners individually prepared a frequency table within five minutes (ref Figure 4.5.2a).
knowledge of how to construct a frequency table
from their previous lesson as they prepared it
efficiently (line 2 and table 4.5.2a).
97
Table 4.5.2b: Frequency table showing the performance of learners in a test
Scores (x)
Tally
Freq. (f)
Fx
2
/
1
2
3
/
1
3
4
//
2
8
5
///
3
15
6
//
2
12
7
/
1
7
∑f = 10
∑fx = 47
Construction of a bar graph
Teacher content knowledge was used to describe
Line 3a: Teacher B described algorithmically how to construct a single bar graph, using the data from the frequency
how a bar graph is constructed (lines 3a and 4).
table (ref Figure 4.5.2a) prepared by the learners as indicated in line 2. ‘Now, watch out, you begin by drawing the
Teacher B probes learners with a question to find out
vertical and horizontal axes’ he said. Teacher B drew the horizontal and vertical axes and asked the learners to
if they know how to choose a scale for constructing a
explain how to choose the scales for the axes. He asked, ‘How do we choose the scale for labelling the axes?’
bar graph (line 3a).
Line 3b: Some learners raised their hands and the teacher pointed at one to explain.
Line 3c: Learners stated how the numbers should be written on both the horizontal and vertical axes, by indicating 1,
2, 3, 4 etc for the horizontal axis and 2, 3, 4, 5, 6, and 7 on the vertical axis, while the teacher wrote the numerals on
the chalkboard and elucidated why the scale had been accepted for constructing the bar graph, for such reasons as
considering the highest and lowest values on the frequency table and data and the dimension of the graph paper.
Teacher B merely indicated the scale that was chosen
by the learners and why it accepted and wrote them
on the chalk board with no mathematical justification
of how either the learners or himself had selected the
scale (line 3a)
98
Line 4a: Teacher B showed how to label and draw the bars, using the appropriate frequencies on table 4.5.2a:
Teacher content knowledge was used to describe
’Watch and see how to draw the bars; the first score is 2, and the corresponding frequency is 1’, the teacher said.
how to construct a bar graph using a procedural
Learners watched as the teacher demonstrated how to draw one of the bars on the axes corresponding to the score
approach (instructional strategy) (lines 3a, 3c, 4a,
(data axis) with a value of 2 and frequency is 1.
4c and 5).
Teacher B analysed learners’ classwork as he
monitored their work on bar graphs (line 4c).
Graph construction skills (drawing the axes,
choosing scales, labelling axes, plotting the points
and drawing the lines of best fit) were used by
learners in drawing a bar graph (lines 3, 4a and 4b,
and Figure 4.5.2a) (instructional skill).
Misconceptions
Figure 4.5.2a
Bar graph of the scores of learners in test on how to construct, analyse, and interpret a bar
and
learning
difficulties
in
constructing a histogram instead of a bar graph
were identified by monitoring and analysing learners’
graph using the scores in column 1
responses to classwork and in the class discussion
Line 4b: The teacher asked, ’How can I draw the second bar graph?’ The teacher nominated one of the learners, who
answered, ‘The second score is 3, and the frequency is 1’. The teacher demonstrated how to draw the second bar
(indicating that he was satisfied with how the first bar was constructed) and instructed the learners to copy and
complete the bar graph in their exercise books while he monitored them. While monitoring, he discovered that
certain learners experienced some difficulties because they had not left a constant space between the bars, which he
indicated without explanation. He intervened by helping the learners to complete the bar graph and indicated that
there should be constant spacing between the bars.
99
(lines 4c and 4d). Learners may have experienced
such difficulties due to insufficient explanation of
why there should be constant spacing between the
bars of a bar graph (line 4b).
Line 4c: He asked the learners to watch while he completed the bar graph on the board. Learners who were
experiencing learning difficulties (e.g. constructing a bar graph like a histogram) corrected their mistakes as he did so
(see Figure 4.5.2a). Line 4d: The learners asked, ‘Why they had to leave spaces between the bars?’
Line 5: Teacher B referred to the graph on the chalkboard and answered: ’The bars represent different scores; the
Teacher B answered the learners’ question by
height of the bars represents the number of learners that scored a particular mark, e.g. two learners scored 4 marks,
demonstrating how to label the axes and explaining
and the constant spacing differentiates one score from another, as the number of learners that score a particular
why it is necessary to leave constant spaces between
mark is not the same’
the bars (line 5) (teacher content knowledge).
Line 6: Learners were given time to correct their misconceptions in their notebooks, as well as learning difficulties.
Teacher B used the instructional strategy of again
Afterwards, the teacher explained again how to construct the frequency table and bar graph as he did in line 4a to 4c,
explaining the preparation of a frequency table and
as some of the learners continued to ask for clarity on why there should be constant spacing between the bars.
bar graph construction to clarify learners’
understanding of the need for constant spacing
between the bars of a bar graph (line 6).
Line 7: The learners asked: ‘How do you know that the 10-mark test was easy or difficult?’ This question demanded
Learners asked a question that required the teacher to
that the teacher explain the relevance of frequency tables and bar graphs, which he had not done initially.
explain the relevance of frequency table and bar
graphs (line 7).
Line 8: Teacher B explained: ‘Other factors could be used to determine whether the test is easy or difficult, but at
Teacher content knowledge was used to explain the
the moment, the pass mark is considered’. For example, ‘If the pass mark is 4 and the number of learners that scored
criteria and demonstrate how to determine whether
4 and above is 8 out of 10 learners, then the test was easy’. Teacher B read out the number of persons who scored 4
the 10-mark test was difficult or easy (line 8)
and above as 8. ‘This means that about 90% of the learners scored between 4 and 10. But if the number of learners
(teacher content knowledge).
that scored between 1 and 3 is 8 (Teacher B read from the graph), and the highest score was 5, the test was difficult,
as 80% of the learners scored below 4 marks’. He continued, ‘Thus, with a bar graph, it is easy to show and
interpret learners’ performance in a test. From Figure 4.5.2a, it is evident that the test was within the level of the
learners, as the learners’ marks were not too low, and if the pass mark was 4 (40%), then only two of the learners
failed’.
100
Line 9: Teacher B gave out photocopies of classwork, in which learners were asked to construct a bar graph
Iindividual learners did classwork on bar graphs
individually. The teacher monitored and analysed their responses as they worked. Some learners had drawn their
efficiently (independent instructional strategy)
diagrams, but had failed to consider the concept of equal spacing (maybe the learners had not understood the
(line 9).
teacher’s earlier explanation of how and why to leave constant spacing between the bars), causing them to construct
bar graphs that resembled histograms. ‘The spaces between the bars and the width of each bar should be the same to
differentiate one item from the other, although the height of the bars will be different, because of differences in
Learning difficulties occurred from misconceptions
(constructing a histogram instead of a bar graph) (line
9).
frequency’, the teacher said as a way of correcting the learning difficulty during the lesson.
Learning difficulties were identified through
analysis of their classwork (line 9).
Line 10: Teacher B attempted to correct the misconceptions by explaining again how to construct the bar graph on
Instructional strategy of again demonstrating how
the chalkboard, while the learners watched. The teacher again demonstrated how the axes were drawn, followed by
to construct a bar graph was used to correct learners’
choosing the scale, labelling the axes and drawing the bars. The problem arose because the teacher had not explained
misconceptions (line10). The difficulties that the
the reasons for the spaces between bars at the beginning.
learners experienced could be traceable to insufficient
explanation of how to construct a graph using a
procedural knowledge approach (line10)
Line 11: Teacher B provided additional problem-solving activities based on familiar situations (ref table 4.5.2c). For
Problems related to a familiar situation were used by
example, learners were provided with a table containing the amount spent on groceries purchased from a
Teacher B to try to address the learning difficulty of
supermarket, and were asked to draw a bar graph and to determine what percentage, of the total amount spent, the
drawing a histogram instead of a bar graph (line 11).
most expensive item constituted.
Table 4.5.2c:
Frequency table showing the distribution of the amount spent on buying some groceries from
a supermarket
Item
Tomatoes
Rice
Chicken
Maize meal
Onions
Amount
R10
R70
R35
R42
R3
101
Table 4.5.2b contains items bought in a supermarket and the amount spent on each. For example, R10 was spent on
buying tomatoes, R35 on buying chicken, etc.
Line 12: Using table 4.5.2b, some of the learners tried to construct the bar graph quickly and efficiently, beginning
Learners showed evidence of having understood the
with the labelling of the axes, choosing the scale for drawing the bar graph, labelling the vertical and horizontal axes,
lesson on the construction of bar graph (line 12)
plotting points and drawing the bars, but a few still experienced certain difficulties, as they continued to ask why
(construction skills of drawing the axes, labelling
each bar should be separated from the other. This might indicate that either the learners lack the ability to understand
axes, choosing scale, plotting the points and drawing
or that the teacher’s explanation was not sufficient to elicit an understanding of what had been explained.
the line of best fit). Some learners continued to
experience difficulties despite the teacher’s further
explanation of bar graph construction (line 12). The
teachers’ explanation may not have sufficiently
helped the learners to grasp what he had taught or the
learners lacked the ability to understand the
explanation (line 12).
Line 13: The lesson concluded with oral questioning. For example, the teacher asked, ‘Why do we separate one bar
Teacher B asked oral questions and gave homework
from the other with a space?’ Homework on the construction and interpretation of bar graphs from their textbook
to learners on construction and interpretation of bar
was also given to the learners to reinforce their understanding of the construction of bar graphs. Teacher B promised
graphs to reinforce their understanding (line 13).
to use extra tutoring to help learners who were still experiencing difficulties.
Line 14: A post-teaching discussion took place after the lesson in which some of the learners sought clarity on how
Post-teaching discussion was used to address
to calculate the percentage of the most expensive items bought in the supermarket, which was one of the questions
learners’ questions (line 14).
that had not been answered from the classwork. The teacher had to explain orally and asked the learners to complete
it at home.
CLASSROOM PRACTICE (SECOND LESSON OBSERVATION)
Topic: Construction, analysis, and interpretation of ogives. Class: Grade 11
102
Line 1: Teacher B, standing in front of the class, introduces the lesson ‘Today’s lesson is about the construction and
The lesson was introduced by oral probing
the interpretation of ogives’ Oral questions were directed at individual learners as in line 2.
questioning (instructional strategy) (lines 2 and 4)
to identify learners’ conceptions (preconception)
Line 2: The teacher pointed to individual learners and asked them to mention ways in which data may be represented.
Teacher B identified learners who would answer the
question (line 2).
(instructional strategy).
Line 3: The learners referred to the frequency table, the bar graph, the pie chart, the histogram, the line graph, etc.
The learners’ responses to the oral probing showed
that they had insight into how to represent data (line
3)
(content knowledge).
Line 4: Learners were referred to page 199 of their textbooks, activity 8.11, question 3, which contains the mark
Instructional strategy of assessing how to construct
distribution of learners’ performance in an English examination. The teacher requested the learners to:’ (a) prepare a
and interpret an ogive was set for the learners and to
cumulative frequency table of the learners’ performance; (b) construct an ogive; (c) interpret the ogive by
be used to demonstrate how to do so (line 4).
calculating the five-number summary (minimum, first quartile (Q1), median (Q2), third quartile (Q3) and maximum
value’. Although question was set for the learners, but the teacher has to use it as an example to demonstrate how to
construct and interpret ogive.
Teacher content knowledge was used to prepare a
a) Preparation of cumulative frequency table
Line 5a: Teacher B demonstrated how a cumulative frequency table is constructed (see table 4.5.2c), using the first
cumulative frequency table (line 5).
three rows of the table, and said ‘add the frequency of the first and second rows to give the cumulative frequency of
Instructional strategy to assess learners’
the second row (0 + 2 = 2 of second row). The cumulative frequency of the second row is added to the frequency of
understanding of a cumulative frequency table took
the third row to give the cumulative frequency of third row (2 + 6 = 8 of the third row), and so on’ (table 4.5.2c). He
the form of group work activities in class
added, ‘In groups of eight, complete the table by calculating the cumulative frequencies of the remaining intervals
(interactive instruction) (line 4), (line 6b).
within 10 minutes.’
Teacher’s procedural knowledge was used to
demonstrate how to prepare a frequency table (line
5a).
103
Table 4.5.2d:
Mark distribution of learners in an English examination
Marks
Freq (f)
Cumulative
frequency
1–10
0
0
11–20
2
0+2=2
21–30
6
2+6=8
31–40
7
8 + 7 = 15
41–50
14
15 + 14 = 29
51–60
20
29 + 20 = 49
61–70
35
49 + 35 = 84
71–80
29
84 + 29 = 113
81–90
6
113 + 6 = 119
91–100
1
119 + 1 = 120
Line 5b: The learners completed the cumulative frequency table.
Instructional skill mostly used in constructing an
a) Construction of ogive
Line 6a: Teacher B explained procedurally ‘An ogive is constructed by drawing and labelling the axes with data on
ogive was a topic-specific construction skill (lines
the horizontal axis and the cumulative frequencies on the vertical axis. The cumulative frequencies will help in the
6a and 6bi).
construction of the ogive’ he said.
A procedural knowledge approach was used
(content knowledge and instructional strategy) to
demonstrate how to construct an ogive (line 6a and
6bi).
Teacher content knowledge was utilised to provide
104
descriptions on how to plot the points from the
frequency table on the axes of the ogive (teacher
content and instructional strategy) (lines 6a and
6bi).
Interpretation of ogive by calculating the fivenumber-summary (minimum value, Q1, Q2, Q3 and
maximum values) (line 6bii) was carried out by
Teacher B.
Figure 4.5.2b:
a)
Ogive representing learners’ performance in an English examination
Line 6bi: The teacher demonstrated how to construct an ogive by plotting the cumulative frequency against
the marks (e.g. 10, 0; 20, 2; and 30, 8) as indicated on the frequency table and joining the line of best fit.
Afterwards the analysis and interpretation were performed using the formula for calculating the quartile
positions and the quartiles.
Line 6bii: The teacher showed the learners how to calculate the position of the quartiles and said ‘Using the formula
Teacher B provided insufficient explanation (PCK).
n +1
n +1
th , you can calculate Q1 position and locate Q1. Using the formula
th , you can calculate Q2 position
4
2
He focused on procedure at the expense of conceptual
and locate Q2. Using the formula
3( n + 1)
th , you can calculate Q3.and locate Q3.’ All answers were obtained by
4
understanding. Hence learners were obliged to
request further clarification about the position of the
cumulative frequency on the ogive, which the teacher
using the position of the quartile calculated to read out the values of the five-number summary, such as minimum
had not previously explained (line 6c) (teacher
value = 10, Q1 = 52, Q2 = 63, Q3 = 73 and maximum value is 120, from the ogive.
content knowledge and instructional strategies).
Line 6c: A learner asked, ‘Must the cumulative frequency always be on the vertical axis? Why don’t you put it on the
horizontal axis?’ This question showed that the learner did not understand how to label the axes of the ogive because
105
the teacher had not explained this from the beginning, depicting the fact that the teacher displayed insufficient
content and pedagogical knowledge to demonstrate how to label the axes of an ogive.
Line 6d: Teacher B responded, ‘You can label it on the horizontal axis, but it is more convenient to label it on the
vertical axis, as you are expected to plot the cumulative frequency against the marks’ (see Figure 4.5.2b).
Line 7a: Teacher B referred the learners to a photocopied exercise for classwork with a similar question in which
Teacher content knowledge was used to set
learners were requested to prepare a frequency table, construct an ogive with the table prepared and calculate the
classwork on ogive construction (line 7a) to ascertain
five-number- summary (min, Q1, Q2, Q3 and maximum values) from the ogive, but with class intervals starting from
how well learners have understood the lesson.
20. He monitored them while they were doing their classwork.
Line 7b: Most learners misunderstood the concept of labelling class intervals 0–10, 11–20, 21–30, and 31–40, etc.
Instead, they labelled the class intervals on the horizontal axis 20–30, 30–40, 40–50, and 50–60, etc, instead of 10,
20, 30, etc. This approach does not allow the learners to plot the points on the data axis.
Learners’ misconceptions (line 7b) involving how
to label the horizontal axis were identified through
analysis of their classwork. The labelling could
result in drawing a histogram instead of an ogive.
Lack of understanding stemmed from insufficient
elucidation, focusing on the procedural knowledge
approach at the expense of conceptual knowledge
(line 7b) (Learning difficulty)
Line 8a: The lack of understanding of how to label the axes was addressed by Teacher B in a class question and
A rule-oriented procedural approach was used to
answer session (see line 8b). He also explained again how to construct an ogive, as in line 6a, and interpret the ogive,
re-explain ogive construction (line 8b).
as in line 5a.
Teacher B explained once more how to construct
Line 8b: Teacher B referred the learners to the diagram on the chalkboard (Figure 4.5.2b). He again explained how
the ogive was interpreted by means of quartiles by using the formula Q1 =
1
( n + 1)th for the first quartile, Q2 =
4
106
ogive and position of quartiles to reinforce learners’
understanding of ogive construction and
interpretation (line 8a)
(
n +1
)th for the second quartile, and Q3 = ¾(n + 1) for the third quartile, to calculate the first, second, and third
2
quartile position and the first, second and third quartile. These quartiles were used to interpret the ogive by deciding
the percentage of learners who passed or failed the examination by gaining a given pass mark such as the median.
Line 9: The other strategy used to address the learning difficulties was the provision of extra-class activities in their
Extra-class activities on ogive construction were
textbooks for the learners to solve after normal school hours. Its focus was on drill and practice, using the exercises
given
from their textbooks, in order to make the lesson more accessible and comprehensible to the learners.
(instructional strategy) (line 9) to deepen their
to
the
learners
from
their
textbook
understanding of ogive construction and address
learning difficulties.
Line 10a: Teacher B concluded the lesson with oral questioning. For instance, Teacher B asked, ‘What does ‘n’
Oral questioning was used in addition to monitoring
represent in the formula for calculating the quartiles? Where can I locate the quartiles using the formula?’ Teacher
classwork and homework to assess how well learners
B nominated learners to answer the questions after many of them raised their hands.
had achieved the learning outcomes of the lesson
(line 10a). The intention of continuous learner
assessment is to ascertain how well learners have
Line 10b: A learner answered the first question by saying ’n = 120 (meaning the sum of the frequencies as in the
understood the teacher’s elucidation of ogive
diagram on the chalkboard)’. A second learner indicated the answer on the vertical axis but got it wrong. A third
construction during the lesson. (Teacher topic-
learner explained, ‘you have to trace it through the vertical axis to meet the curve, and then go down to the
specific
horizontal axis, where you have to read off the value for the quartiles, e.g. Q1 = 52’ The teacher and learners
knowledge were used to determine learners
accepted the answer.
progress) (line 10a).
Line 11: Teacher B gave the learners homework by referring them to the same exercise in their textbook as
Homework was used as an instructional strategy to
mentioned in line 4, as well as to photocopies of past question papers containing questions related to the
assess how well learners understood the lesson on
construction, analysis, and interpretation of ogives.
ogives and consolidate the lesson (line 11).
107
content
knowledge
and
pedagogical
Line 12: After the lesson, some learners asked him how to label the horizontal axis if the class boundaries did not
Post-teaching discussion took place between the
start from zero. The teacher explained once more to the learners one by one using the example that had previously
teacher and the learners immediately after the lesson
been given in class.
to address the learning difficulty (line 12) (teacher
content knowledge and instructional strategy).
108
Summary of lesson observation of Teacher B
From the two lessons observed, it is evident that Teacher B demonstrated his knowledge of
the content of school statistics which may have been developed through formal education and
teaching with the recommended textbooks and work schedule. Teacher B used appropriate
topic-specific instructional skills and strategies, such as the use of examples drawn from
familiar situations and a formal procedural approach in teaching the construction of the bar
graph and ogive. In statistical graph construction and interpretation, measures of central
tendency, knowledge of graphing involving drawing axes, choosing scale, etc, are regarded as
prior knowledge. In order to identify learners’ preconceptions in bar graph and ogive
construction, he applied diagnostic techniques of pre-activity that focused on the preparation
of a frequency table of ungrouped data and oral questioning on different ways of representing
data. The learners displayed evidence of possessing previous knowledge of bar graphs and
ogive constructions but with no preconception identified, depicting the fact the teacher may
not have had sufficient knowledge of the learners’ likely preconceptions of bar graphs and
ogives.
The learners’ misconceptions in drawing a histogram instead a bar graph, and the learning
difficulties that emanated from these, were identified through analysis of their classwork
while monitoring, checking and marking their responses to the tasks. Further explanations,
extra-class activities and post-teaching discussion were provided to correct their
misconceptions and learning difficulties. Teacher B’s PCK is largely procedural, focusing on
rules and algorithms, and is not always responsive to the needs of the learners, especially
when these involve clarification of the construction of grouped data (the ogive). The frequent
use of procedural knowledge may stem from the nature of the topic, which requires learners
to collect, organise, construct, analyse, interpret statistical and probability model to solve
related problems (DoBE, 2010) and demonstrate how graphs should be constructed
(Leinhardt et al, 1990). This approach did not appear to accommodate the needs of the
learners, because most of them still experienced difficulties with labelling the data axes of
graphs of grouped data. Teacher B can be said to have displayed insufficient ability to
elucidate concepts of ogive construction (PCK), focusing on procedural, at the expense of
conceptual understanding.
109
4.5.3
School C: Lesson observation of Teacher C
In this section, the teacher’s classroom practice on teaching the construction of ogives and
scatter plots is described. The condition of the classroom is described first, followed by his
classroom practice in the construction, analysis, and interpretation of ogives and scatter plots.
110
Table 4.5.3a: Description of lesson observation and classroom conditions in Teacher C’s mathematics lesson
DESCRIPTION OF LESSONS
CATEGORISATION/THEMES
Condition of the classroom
1) The classroom was conducive for learning, safe and
Teacher C’s classroom was safe and protected. The teacher had a full view of the entire class during lessons. The
well protected.
classroom walls were decorated with science wall charts; the furniture, windows and door were in good condition,
2) There were 45 learners in the class, who were seated
with electrical wiring that permitted the use of appliances such as an overhead projector. The individual learners
in double chairs in four columns.
were resourced with textbooks, calculators, exercise books, and graph sheets for each learner, as well as
.3) The individual learners have all the necessary
construction instruments for the teacher (ruler, protractor, and pair of dividers).
materials for learning statistical graphs.
There were 45 learners, consisting of 26 females and 19 males, seated comfortably in twos in four columns of
double chairs and desks.
CLASSROOM PRACTICE (FIRST LESSON OBSERVATION)
CATEGORISATION/THEMES
Topic: Construction, analysis, and interpretation of ogives. Class: Grade 11
Line 1: A histogram had been taught in the previous lesson, and learners had been given homework.
The ogive was taught (line 1). (Teacher content
knowledge).
Line 2a: Teacher C and the learners marked the homework on the construction, analysis, and interpretation of the
Oral probing questioning to identify learners’
histogram.
conceptions (preconceptions) (lines 2bi and 2c) was
used to introduce the lesson (Instructional strategy).
Line 2bi: To determine learners’ prior knowledge of ogives, Teacher C asked, ‘What is the difference between a
class interval and a class boundary?’
Line 2bii: One of the learners voluntarily answered, ‘A class interval and a class boundary are the same thing,
111
Analysis of homework (checking if answers were right
or wrong) (line 2a) was used to try to identify learners’
because both of them contain a group of numbers between them.’ The question was not answered correctly, but
conceptions in ogive construction (instructional
none of the other learners volunteered to answer. Other learners, Teacher C indicated, could not provide the
strategy).
answer. Therefore, the teacher explained, using an example, ‘0–10, 11–20, 21–30, etc., are class intervals. But 0–
10, 10–20, 20–30, etc’ are class boundaries of a prepared ogive on a photocopied exercise.’
Teacher content knowledge was used to explain the
Line 2c: Teacher C requested. ‘indicate to me how data can be represented based on your experience’
Line 2d: Learners referred to the bar chart, the pie chart, scatter plots, the line graph, ogive, etc. This response
differences between class boundaries and intervals (line
2bii).
indicated that learners held some conceptions about ogives, which included data representation, since they had
The learners displayed evidence of having previous
been taught previously.
knowledge about data representation in statistics (line
2c)
Line 3a: Teacher C explained the construction of the ogive procedurally, using a frequency table on a photocopied
Teacher content knowledge was used to set the
exercise containing the ages of cars, in years, in a sample of 100 car owners. Learners were also asked to interpret
example to demonstrate histogram construction (line
the ogive in terms of the five-number-summary. A five-number-summary consists of the minimum value, Q1, Q2,
3a).
Q3, and Maximum value of the given data.
Line 3b: A cumulative frequency distribution table was individually constructed by the learners, based on the
teacher’s instruction (ref Table 4.5.3b). For instance, Teacher C explained: ‘The frequency of the first row (25)
should be written under the column for cumulative frequency. The cumulative frequency of the first row is then
added to the frequency of the second row (25 + 32 = 57), to get the cumulative frequency of the second row, etc’.
Learners’ content knowledge was used to complete
the cumulative frequency table in a procedural manner
(instructional strategy) following certain algorithms
(lines 3a and 3b).
Teacher content knowledge in statistics (data
collection) was used to prepare a frequency table (lines
3b and 3c).
112
Table 4.5.3b:
Table showing the ages of cars in a sample of 100 cars
Age (years )
Freq. (f)
Mid-values
(x)
fx
Cum.
freq.
0<x<2
25
1
25
25
skills (drawing of axis, choosing of scale, labelling of
2<x<4
32
3
96
57
axes, plotting the points and joining the line of best fit)
4<x<6
20
5
100
77
(line 3d) were used to construct the ogive.
6 < x < 10
12
8
96
89
10 < x <15
7
12.5
87.5
96
15 < x < 20
4
17.5
70
100
∑f
= 100
Instructional skills such as topic-specific construction
Topic-specific
teacher
content
knowledge
and
instructional strategy were used to demonstrate how
∑ fx = 474.
to construct an ogive (line 3d) using an algorithmic
approach. Thus the teacher has content and pedagogical
Line 3c: ‘Continue in the same way to calculate the remaining cumulative frequencies,’ Teacher C said. The
knowledge of histogram construction.
learners, as shown in table 4.5.3b completed the table.
a) Construction of ogive
Line 3d: Teacher C used Table 4.5.3b to explain how to construct the ogive, as shown in Figure 4.5.3a. He
requested, ‘draw the vertical and horizontal axes, choose a scale by considering the highest and lowest value on the
cumulative frequency and class boundaries’. The teacher used topic-specific algorithmic knowledge of ogive
construction in his demonstration.
113
A procedural knowledge approach was used to
explain how to construct an ogive (lines 3c and 3d)
(instructional skill and strategy)
Figure 4.5.3a:
Ogive of age distribution of sample of 100 cars owners park in a car park
Line 4: Teacher C explained the ogive construction, using a rule-oriented approach, plotted two points and asked
Teacher C’s use of an algorithmic approach to explain
the learners to complete the plotting and join the lines of best fit for the ogive as part of their classwork.
how to construct an ogive (Teacher content
knowledge and instructional strategy) (line 4)
Line 5: Learners completed the ogive by plotting (6; 77), (10; 89), (15; 96) and 20; 100) and joining the line of
Learner content knowledge was used to complete the
best fit. (ref Figure 4.5.3a). But some learners were uncertain about the labelling of the data axis.
ogive (line 5) but some of the learners appeared not to
have understood how the ogive was completed
especially the labelling of the data axis with data from
the frequency table.
Line 6a: Teacher C monitored the learners and offered a further explanation of the preparation of the cumulative
Teacher C monitored and guided learners while they
frequency table to those who were experiencing difficulties, such as being uncertain how to label the data axis with
were doing their classwork (instructional skills and
the class boundaries provided on the table of values. He indicated, ‘The cumulative frequency was used to label the
strategies) (line 6a).
cumulative frequency axis (vertical axis) and data axes on the horizontal axis’.
Insufficient explanation was provided because a
procedural approach was used where a conceptual
114
Line 6b: A learner asked, “Why do we need to add these numbers (frequencies) together?’
explanation was more appropriate (line 6b).
Line 6c: Teacher C answered, ‘Adding the frequencies together to give the next frequency on the cumulative
Teacher content knowledge was used to explain how
frequency column makes it a cumulative frequency that you are required to calculate for constructing the ogive.
the cumulative frequencies were obtained (conceptual
Cumulative means adding more numbers each time to get the next number.’ Through non-verbal cues of nodding
knowledge approach) (line 6c) (instructional
their heads up and down, learners showed that they had understood the explanation, indicating that the conceptual
strategy).
knowledge approach was sufficient to enable them to comprehend how a cumulative frequency table is prepared.
Teacher C demonstrated the required content knowledge of preparing a cumulative frequency table in his
explanation regarding the construction of an ogive to the learners.
Line 7a: Teacher C observed a misconception, which resulted in drawing a histogram instead of an ogive with the
Misconception of drawing a histogram instead of an
given data, while he monitored and analysed the learners’ responses to classwork.
ogive was identified during monitoring of classwork
(line 7a).
.
Line 7b: Teacher C told a learner who was experiencing this misconception, ‘Look, you were asked to complete
the ogive we were plotting on the chalkboard and not to draw something else. Clean it off and continue with the
diagram on the chalkboard by plotting the points and joining the line of best fit. For example, when cumulative
frequency is 57, age is 4; when cumulative frequency is 77, and age is 6; etc,’ the teacher said.
Learning difficulties resulting from this misconception
were identified through analysis of learners’ responses
to classwork (line 7a).
Teacher C addressed the misconception through
reviewing the learners’ work and instructing them to
continue with plotting the points and joining the line of
best fit (line 7b). (Teacher C displayed knowledge of
the topic content, instructional strategy and learning
difficulty.)
115
Line 8a: Referring to how the horizontal axis was labelled, a learner asked, ‘Why do you indicate the numbers that
Teacher C identified lack of knowledge or his
were not on the table?’ The learner displayed a lack of knowledge of selecting a scale of given grouped data,
insufficient explanation (learning difficulty) of how to
which may not have been addressed through the procedural approach adopted by the teacher.
choose a scale for constructing an ogive through oral
questioning from the learners (line 8a).
Line 9a: Teacher C re-explained the construction of an ogive by analysing the table of values of the cars and how
Teacher content-specific knowledge of the
they were used to construct the ogive, as in line 5. He explained, ‘The numbers were not omitted , but grouped
construction of an ogive was used to explain how to
together as: 0 < x < 2; 2 < x < 4; 4 < x < 6; etc. And 6 < x < 10 contains 6 < x < 8; 8 < x < 10, In addition, 10 <
label the horizontal axis (line 9a).
x < 12, 12 < x < 14, 14 < x < 16, 16 < x < 18, 18 < x < 20 is within 10 < x < 15 and 15 < x < 20, as indicated in
A conceptual knowledge approach based on
the diagram. Indicating those numbers that were not on the table ensured sequential numbering of the data axis
teacher’s content specific knowledge of how to label
that could help in the construction and interpretation of the ogive’.
graphs of grouped data was used to explain the
Line 9b: After plotting the points, Teacher C demonstrated how to join the line of best fit, which gave an S shape.
construction of an ogive (lines 9a and 9b)
He instructed learners to copy the description from the chalkboard.
(instructional strategy).
Line 10: Learners listened, and copied notes from the board. One asked, ‘Does it mean that the graph of the ogive
This question showed lack of understanding of the
must be in the form of an S?’
nature of an ogive. It required further clarification from
the teacher from his content knowledge of ogive
construction using a conceptual knowledge approach
(line 10).
Line 11: Teacher C answered, ‘Yes.’ He explained, ‘ogive graphs are typically in an S shaped. If the constructed
Teacher C answered learners’ oral questions and
graph does not display this shape, then it is not an ogive or is constructed wrongly’.
provided greater clarification to reinforce
comprehension of the nature of an ogive (teacher
content knowledge) (line 11).
The teacher asked how the median is calculated from
b) Interpretation of ogive (calculating the quartiles from an ogive)
grouped data as a way of determining learners’
116
Line 12: Teacher C posed this question to the learners, ‘How would you calculate the median from the ogive,
conception in ogive interpretation (line 12).
according to the question?’
Line 13a: A learner (pointed out by Teacher C) answered, ‘you have to arrange the data in ascending order and
The learners showed lack of comprehension of how to
locate the middle number. But if they are more than one number at the middle, the average of the two middle
calculate the median from a graph of grouped data (line
numbers is considered as the median.’ The learner quoted the wrong formula for finding the median of ungrouped
13)
data, instead of quoting the formula for finding the second quartile of a grouped data showing a lack of
understanding of how to calculate median of grouped data. Line 13b: Teacher C explained the formula for
calculating all the quartiles and focused on the formula for calculating the median position by indicating, ‘Median
(second quartile)( Q2) = (
n +1
)th ). The position of the median calculated (second quartile) was used to locate
2
the median on the ogive. ‘Median age = 3 years’, the teacher said.
An algorithmic approach was used, in that the
quartiles were calculated according to a particular
procedure or formula, without explanation of the use of
that algorithm (insufficient knowledge of learners’
Line 13c: Teacher C and the learners calculated the first and third quartile from the ogive using the formulae (Q1
1
3( n + 1)th
to locate Q1 and Q3. The five number summary was i) minimum age = 1year;
= ( n + 1)th and Q3 =
4
4
conceptions and learning difficulties) in calculating
the median of grouped data, and the difference between
calculating the medians of grouped and ungrouped data)
(line 13a).
Q1 = 2 years; Q2 = 3 years; Q3 = 8 years and the maximum age = 20 years. These were all calculated and listed.
But some learners appeared to be confused because they regarded the quartile position as the quartile itself. For
Procedural knowledge was used to explain how to
example, the first quartile position was calculated as 25.5th position. Rather than using this position to find the
calculate the quartile’s position and locate the quartile
th
value of first quartile from the data, the learners simply wrote Q1= 25,5 instead of Q1= 2. Some learners
displayed a lack of understanding of how to calculate quartiles from the ogive due to the teachers’ procedural
knowledge description of how to calculate quartiles.
itself from the ogive (lines 13b and 13c).
Learners experience some difficulties of using the
quartile position to represent the quartile itself (line
13c) which may be linked to the procedural knowledge
description adopted by Teacher C during the lesson on
ogive construction (line 13c).
117
Line 14: The teacher provided the following detailed explanation of the mathematical connections between the
Teacher content knowledge was used to show the
quartile positions and how they were used to calculate the quartiles from the ogive. The teacher first explained,
mathematical connections between the quartile position,
‘the meaning of ‘n’ is the number of cars in the park. The value of ‘n’ was obtained from the table by calculating
the quartiles and how they are utilised in interpreting
the frequencies, and substituting the value of ‘n’ into the formula (in line 13b and c), you can determine the first
the ogive (line 14) employing a conceptual knowledge
quartile position (Q1)’. His next step was to show the mathematical connection between the quartiles position and
approach.
the value of the quartile from the ogive by using the quartile positions to locate the values of the quartiles from the
ogive as indicated in line 13c. Following his explanation in which he substituted ‘n’ into the formulae as indicated
in line 13b, the quartile positions were calculated and used to locate the values Q1 = 2 years; Q2 = 3 years; Q3 = 8
More learners understood the explanation given via a
years, from the ogive. The learners were able to use the same formula and procedure to calculate the quartile
conceptual knowledge approach (line 14).
positions and the quartiles in their classwork based on the teachers’ conceptual explanation.
Line 15: Individualised teaching took the form of post-teaching discussion, so that each learner presented the areas
The instructional strategy of using more activities
in which he or she was still experiencing problems. The difficulties included labelling data axes and determining
applicable to familiar situations from their mathematics
the median value of an ogive. The teacher provided more activities applicable to familiar situations using their
textbook was used to address learners’ learning
mathematics textbook as a way of reinforcing learners’ competency in ogive construction.
difficulties in labelling the data axes of grouped data
and determining the median of an ogive (line 15)
(knowledge of learners’ learning difficulties and
instructional strategy).
Line 16: The mathematics textbook, as well as examination aids and publications of Study mate containing past
Textbook and other materials were used as sources of
questions in statistics and mathematics, were used by Teacher C to prepare and teach the construction and
information for teaching ogive construction
interpretation of the ogive, as well as to assign homework.
(development of teacher’s PCK in respect of content
knowledge and instructional strategies) (lines 4 and
16).
118
CLASSROOM PRACTICE (SECOND LESSON OBSERVATION)
CATEGORISATION/THEMES
Topic: Construction and interpretation of scatter plots. Class: Grade 11
Line 1: Marking and checking homework on the construction and interpretation of scatter plots was used to start
Teacher C used the instructional strategy of checking
the lesson and to identify learners’ knowledge or conceptions about scatter plot construction after Teacher C had
learners’ homework on scatter plot construction and
greeted the class. After the marking and checking were concluded, Teacher C gave the correct answers, while the
interpretation to try to identify their knowledge and
learners wrote down the corrections in their notebooks.
preconceptions of scatter plot construction (line 1).
Line 2: Teacher C wrote the topic, ‘Construction and interpretation of scatter plots’ on the chalkboard and
Teacher content knowledge of scatter plots was
presented a photocopied exercise containing different types of scatter diagrams to the learners.
utilised to indicate the topic of the lesson and set
activities to ascertain learners’ knowledge of scatter
plot constructions (line 2)
Line 3a: The learners were asked to work in groups and to determine (by analysis and interpretation of the scatter
Learners worked in groups (instructional strategy) to
plots) which of the scatter diagrams had a positive correlation, a negative correlation, or no correlation. They had
analyse and interpret scatter plots as a way of
previously been taught how to construct a scatter plot.
identifying how well they had grasped how to construct
Line 3b: Learners worked in groups to analyse the scatter plots, to determine the nature of the points plotted and
a scatter plot from their previous lesson (lines 3a and
3b).
the lines of best fit.
Line 4a: After the analysis, learners (in groups) were asked to interpret the graph by indicating their conclusions:
Learner activity on data handling and interpretation by
whether the diagrams showed a positive correlation, a negative correlation, or no correlation.
responding to class activities was undertaken in groups
Line 4bi: Learners through their spokespersons for each group indicated, ‘The first diagram displays a positive
(Line 4bi).
linear relationship.’ Another group concluded, ‘the second diagram displays a graph of negative relationship, but
Teacher instructional strategy of giving and
not linear.’ Some of the groups did not seem to be satisfied with the answers presented for two of the graphs B and
monitoring classwork on scatter graph interpretation
C.
was used to identify learner knowledge and conceptions
119
Line 4bii: Teacher C monitored the way in which learners were analysing and interpreting the scatter plots in
of scatter plots (line 4a).
groups. ‘In terms of analysis, you were expected to know the values of Y and the corresponding value of X as used
Learners misinterpreted a scatter plot owing to
in constructing the scatter plots,’ he said. He continued, ‘Based on the relationship between X and Y values, one
insufficient comprehension of scatter plot construction
can say whether there is positive correlation, negative correlation, or no correlation (interpretation) as previously
as a result of inadequate teacher explanations regarding
explained.’ Recognising that some learners appeared to be experiencing difficulties in interpreting a negatively
how to determine the relationship between X and Y in a
correlated scatter plot as having no correlation in interpreting the diagrams, which could indicate that they lack an
scatter plot (learning difficulty) (Line 4bii). A
understanding or the teachers’ previous lesson explanation on scatter plot construction was not sufficient to enable
negatively correlated scatter plot was interpreted as
them to grasp what he had taught them on the topic, Teacher C further handed out another photocopied exercise
having no correlation due to an outlier.
showing a table of values reflecting the age and mass distribution of players in a rugby game. He asked one of the
learners (who appeared to have interpreted the diagram more efficiently), ’Plot the numbers of players against the
masses to construct a scatter plot. Can I see you do that on the chalkboard?’ The learners constructed the scatter
plot efficiently. But Teacher C decided again to assess learners’ conceptions in scatter plot construction (using
Teacher content knowledge was used to explain
extra-class activity) which would have aided them in interpreting the scatter plot if they had known how to
(instructional strategy) the construction and
construct these efficiently. Teacher C used his topic-specific content and pedagogical knowledge to assess the
interpretation of a scatter plot (lines 4di and 4dii).
learners’ understanding of scatter plots using more activities on their construction in order to improve their grasp
of the latter. In this activity, Teacher C plotted some points using the frequency table that he has provided on the
activity on the scatter plot and requested learners to complete the remaining points. He said, ‘let someone complete
the scatter plot?’
A procedural approach of drawing the axes, choosing
Line 4c: More learners volunteered and they were requested individually to plot other points on the graph using the
table provided by the teacher on the chalkboard, while the other learners watched.
scale, labelling axes, plotting the points and drawing the
line of best fit was used to describe and complete the
scatter plot (line 4di).
Line 4di: Teacher C completed the graph that the learners had been plotting, and explained algorithmically how to
construct a scatter plot. He then analysed it by reading the value on the vertical axis and the corresponding value
on the horizontal or data axis. ‘From this analysis, the meaning of what the graph intended to convey about the
120
Graph construction skills (drawing axes, choosing
scale, labelling axes, plotting the points and joining the
relationship between the number of players and their masses (correlation or no correlation) was determined’, the
line of best fit) were used to create a scatter plot (line
teacher said.
4di).
Line 4dii: Some of the learners seemed dissatisfied, as they shook their heads. More explanations were offered by
Teacher C, who utilised a conceptual approach to again demonstrate scatter plot construction and interpretation
Teacher
using the classwork. For instance, Teacher C explained; ‘The characteristics (nature of points and shape of line of
conceptual
best fit) of a linear positive correlation with its line of best fit moves from right to left through the origin, and
difficulties, showing that he has insight into learners’
related it to diagrams A and E of Figure 4.5.3b. In a linear negative correlation the line of best fit drops down
learning difficulties,; hence the strategy he adopted to
from the vertical axis to the horizontal axis, as in diagrams B and C, Figure 4.5.3b. And a scatter plot with no
provide clarification and reinforce understanding (line
correlation has all the points spread through the vertical to the horizontal axis as in diagram F, Figure 4.5.3b’.
4dii)
‘Diagram D shows a positive correlation, but it is not linear because the points spread through the origin from
right to left, but not in a straight line,’ the teacher concluded
Diagram A Diagram B Diagram C
Diagram D Diagram E Diagram F
Figure 4.5.3b: Scatter diagrams showing different kinds of correlation between X and Y
121
C
provided
further
knowledge)
to
explanation
address
(using
learners’
Line 5a: A learner asked, ‘Do we need to draw the line to show how the two variables X and Y are correlated?’
This learner’s question displayed a lack of
This question demanded a conceptual explanation which was provided in line 4dii, but the learner may have
understanding of how to construct and interpret scatter
developed certain misconceptions about drawing the line of best fit in a scatter plot from the earlier procedural
plots─precisely because of inadequate explanation,
explanation which led to a lack of understanding of why such a line has to be drawn based on the nature of the
using learned rules to explain. The question is how does
points plotted, to determine the relationship between X and Y’. Another misconception was, ‘There were no lines
the teacher makes the leap from the algorithmic to the
of best fit in Figure 4.5.3b which they had worked on earlier’, the learner indicated. The learner had posed a
conceptually meaningful explanation (line 5a).
legitimate question seeking clarification because the teacher simply did not provide a conceptual explanation for
the different scatter plots as indicated in the introductory exercise for the lesson and in line 4di.
Line 5b: Teacher C answered, ‘Yes’ and repeated what he had said in line 4dii by explaining the characteristics of
scatter plots, how their correlation can be determine and how they relate to each other as in the diagrams in Figure
4.5.3b.
Line 6: Teacher C observed that in the graphs the learners analysed in groups, they misinterpreted diagram C
Teacher content knowledge was used to address
(Figure 4.5.3b. For example, diagram C was interpreted as a graph with no correlation between X and Y, owing to
learners’ misinterpretation of scatter plot (line 6) by
outliers (the point or points that are farthest from the line of best fit). ‘Using one point alone to indicate that
explaining why diagram C could not be adjudged to
diagram C had no correlation may not be adequate as there are other clustered points that would display the
have a negative correlation. A more conceptual
correlation between X and Y,’ the teacher explained. This was a misconception of using the nature and shape of a
explanation was provided of how to describe the
scatter plot with no correlation to interpret a graph of negative linear correlation. In addition, some learners
relationship between X and Y in a scatter plot and
indicated in their exercise book that the line of best fit meant a change in X caused by a change in Y, as in a line
indicate the kind of correlation that the scatter plot is
graph, which means if Y increases, then X increases by the same percentage. ‘Yes, when X increases, Y also
showing (line 6).
increases, which means X and Y are related,’ one of the learners indicated. In a scatter diagram, ‘The line of best
fit only indicates the association or connection between X and Y, as indicated in diagrams A and B,’ the teacher
explained. He continued, ‘And depending on how clustered the points are close to the line of best fit, one can say
that it is strong, moderate of weak correlation.’ As indicated earlier, ’You were expected to analyse and interpret
the scatter plots to determine the relationship between X and Y,’ he emphasised.
Line 7: Teacher C corrected the misconception of using the characteristics of a scatter plot with no correlation to
122
The topic-specific content and instructional strategy
interpret a scatter plot with a negative linear correlation, as well as interpreting a linear scatter plot as if it were a
of providing more examples was used to address the
line graph, as in lines 5 and 6, and diagram C of Figure 4.5.3b. He provided more activities on scatter plots and
learners’ misconceptions concerning outliers and
photocopied activities on their construction and interpretation of scatter plots. For example, he said, ‘In this
interpreting a linear correlated scatter plot as if it were a
exercise, you were required to construct a scatter plot and indicate the relationship between test 1 and test 2 (see
line graph (line 7). Topic-specific content and
Table 4.5.3c below). The data in the frequency table give the marks (out of 20) that 12 learners attained in the two
pedagogical knowledge was utilised to address
tests’.
learners’ misconceptions.
Line 8: Teacher C gave out the classwork as shown below.
Table 4.5.3c:
Frequency table showing the distribution of learners’ performance in two tests
learner
A
B
C
D
E
F
G
H
I
J
K
L
Test 1
10
18
13
7
6
8
5
12
15
15
10
20
Test 2
12
20
11
18
9
6
6
12
13
17
10
19
a)
Draw a scatter plot and describe by means of two examples whether there is a positive or a negative
correlation in the learners’ performance in the tests.
b) How do you account for the outliers, if any?
Line 9: As he monitored the learners’ doing the first classwork, he discovered that some of them did the classwork
Instructional strategy of using real-life context based
efficiently. He gave a second classwork activity involving a frequency table of the age distribution of persons
examples to assess learners’ conceptual understanding
infected with HIV/Aids in two towns. They were to work on their own individually to construct a scatter plot
of the construction and interpretation of scatter plots
showing the relationship between the age distributions of persons infected with HIV/AIDS in the two towns. The
and address their learning difficulties (line 9). Several
objective of using several activities on scatter plots constructions was to identify and correct any difficulties or
class activities were used to reinforce learners’ grasp of
errors related to the construction and interpretation of scatter plots and reinforce learners’ grasp of scatter plot
how to construct and interpret scatter plot (line 9)
construction.
123
Line 10: Learners carried out the exercise individually. A few still experienced difficulties in drawing the line of
An individualised or independent learning
best fit and determining the type of correlation.
strategy/approach was used to evaluate how well
learners had learned the construction of a scatter plot
(line 10).
Line 11: After the classwork, oral questioning, and homework (as in line 8), were made use of by Teacher C to
Oral questioning and the homework assignment
further assess learning. For instance, he asked a learner, ‘What is an outlier?’ ‘An outlier is a data value or point
comprised the instructional strategy used to assess how
that lies apart from the rest of the data’, the learner replied. Teacher C adjudged the learner to be correct and
well learners had grasped the concept of constructing
instructed the learners to answer other questions on the photocopied exercise as homework.
scatter plots (line 11).
Line 12: At the end of the lesson, some learners asked more questions about the work that they did, especially the
Teacher content knowledge and instructional
misinterpretation of a negative linear scatter plot and interpreting the line of best fit in scatter plot as if it were a
strategy was used to clarify the misinterpretation of a
linear algebraic graph. Teacher C held individual discussions with a few learners about diagram C, and asked the
negative linear scatter plot and interpreting the line of
others to see him after school the following day.
best fit as if it is an algebraic linear graph in a postteaching discussion (responding to learners’ oral and
written questions after lesson) and various examples
(line 12).
124
Summary of lesson observation of Teacher C
The way in which Teacher C taught his lessons on the ogive and scatter plot showed that he
possessed the subject matter content knowledge of school statistics. He utilised recommended
statistics and statistics-related textbooks and materials (mathematics study guides) to teach
statistical graphs such as the ogive and scatter plot. He demonstrated his subject matter
content knowledge by describing how the ogive and scatter plot should be constructed, by
adopting an approach that emphasised procedural knowledge and application of formulae,
rather than conceptual knowledge. For example, the teacher made greater use of algorithms
by slotting values into equations for calculating quartiles without eliciting clear
comprehension of the relationships of concepts in the equations. At times he did not provide
adequate explanation, and merely repeated the procedures for arriving at an answer when the
learner experienced misconceptions and learning difficulties in interpreting an ogive using
the calculated quartile positions. Having said that, the teacher used his conceptual knowledge,
for instance on how to teach ogive and scatter plot construction, especially when learners
encountered misconceptions and learning difficulties such as drawing a histogram instead of
an ogive, being unable to label the data axis because of incorrect scaling, and not knowing the
distinction between quartile position and quartile value to teach ogive and scatter plots. While
the teacher used his procedural knowledge to explain in a step-by-step manner how ogive and
scatter plots are constructed, he employed his conceptual knowledge to demonstrate the
mathematical connections between quartile positions and to utilise the calculated quartile
position to work out the quartile value from the ogive in order to provide the meaning or
information that the ogive conveys (interpretation). For example, while the quartile position
for Q1 was calculated to be 25.5th, Q1 value from ogive was found to be, Q1 = 2.
Concerning the instructional knowledge component of his assumed PCK in data handling,
Teacher C used appropriate topic-specific scatter plot construction skills of drawing the axes,
choosing the scale, labelling of axes, plotting the points and joining the lines of best fit to
make data-handling lessons on ogives and scatter plots accessible to more learners. Postactivity and post-teaching discussions were among the instructional strategies he used to
address errors and construction difficulties, etc, in ogives and scatter plots. He applied the
required diagnostic techniques of oral probing / questioning, checking and marking of
homework at the beginning of the lesson to try to identify learners’ prior knowledge about
ogive and scatter plot construction. Teacher C identified learners’ previous knowledge
125
instead of preconceptions which could indicate that the teacher may not have possessed
sufficient knowledge of learners’ preconceptions in ogives and scatter plots constructions.
The lack of sufficient knowledge of learners’ preconception which could have been used to
address any anticipated learning difficulties during lesson planning and implementation may
have further created room for learners to develop some misconceptions and such learning
difficulties as an inability to label data axis, constructing a histogram instead of an ogive and
misinterpreting a negative correlated scatter plot as having no correlation. These
misconceptions in using content knowledge about algebraic line graph construction to
interpret the line of best fit of a scatter plot and learners’ inability to label the data axis were
identified through analysis of their responses to classwork and homework, and pre- and postteaching discussions. Teacher C provided additional class activities and individualised
teaching, post-teaching discussion on the classwork, and further elucidation on scatter plots
immediately after the lesson in order to correct any remaining misconceptions and learning
difficulties.
From the analysis of the lesson observations of Teacher C, it appears that his PCK was more
frequently a procedural approach to teaching, and less often a conceptual approach. The
frequent use of procedural knowledge may be a result of the nature of the topic, which
requires learners to be able to collect, organise, construct, analyse, and interpret statistical and
probability models to solve related problems (DoBE, 2010) and to demonstrate the
construction skills of graphs in statistics (Leinhardt et al, 1990). Following this sequence, the
teacher may have decided to use his procedural knowledge to teach the construction and
interpretation of ogive and scatter plots. On the other hand, the teacher adapted his conceptual
knowledge to explain the construction and interpretation of ogives, especially when learners
experienced misconceptions and learning difficulties. For example, when some of them
misinterpreted a negative linear scatter plot as having no correlation because of an outlier, the
teacher explained the meaning and nature of the scatter plot and its line of best fit, which can
be used to determine the extent of the correlation (strong, moderate, weak or no correlation)
(ref Second lesson observation, line 6). The mathematical connection between calculating the
quartile position and using the calculated position to locate the quartile in an ogive was
explained conceptually to the learners when they could not distinguish between them during
his lesson on ogive construction that involved a procedural approach (line 14).
126
While the teacher can be said to comprehend learners’ learning difficulties by identifying
problem areas through the analysis of learners’ classwork, homework and from pre-and postteaching discussion, as well as addressing the difficulties using familiar context-based
examples, his knowledge of learners’ conceptions may have been developed through the use
of oral questioning, checking and marking of learners’ homework to assess learners’
conceptions in ogive and scatter plot construction.
4.5.4
School D: Lesson observation of Teacher D Grade 11
This section describes briefly the teacher’s classroom practice on the teaching of the
construction of bar graph and histogram. The condition of the classroom is described first. It
is followed by a description of the teachers’ classroom practice in the implementation of the
planned lesson on the construction and interpretation of bar graph and histogram.
127
Topic: Construction, analysis, and interpretation of bar graphs
Table 4.5.4a:
Description of lesson observation and classroom conditions in School D
DESCRIPTION OF LESSONS
CATEGORISATION/THEMES
Condition of the classroom
1) The classroom is safe and conducive to teaching and
There are 17 male and 23 female learners of mixed ability. Forty learners are seated comfortably in twos in four
columns of double chairs and desks. The teacher had a full view of the entire class during lessons. The classroom walls
were decorated with science wall charts; the furniture, windows and door were in good condition, with electrical
wiring that permitted the use of appliances such as an overhead projector. The individual learners were resourced with
learning.
2) The individual learners were resourced with learning
materials.
3) There were forty learners in the class.
textbooks, calculators, exercise books, and graph sheets for each learner, as well as construction instruments for the
teacher (ruler, protractor, and pair of dividers). The classroom presented a conducive learning environment, with locks,
keys, and burglar bars for supervised entry
CLASSROOM PRACTICE (FIRST LESSON OBSERVATION)
CATEGORISATION/THEMES
Topic: Construction and interpretation of bar graphs. Class: Grade 11
Line 1: Teacher D introduced the lesson on bar graphs after greeting the class with a pre-activity exercise in which
Teacher D utilised a learner pre-activity exercise of frequency
learners were asked to individually prepare a frequency table (shown below) of raw data about the number of cars in a
table preparation, which he regards as important for successful
car park manufactured by different companies.
bar graph construction, to try to identify learners’ prior
knowledge or conceptions (preconception) about bar
graphs(line 1) (teacher content specific knowledge and
Table 4.5.4bi: Table showing the number of makes of cars in a car park
instructional strategy)
Company
Nissan
VW
Toyota
BMW
Tata
Number of cars
4
5
8
10
3
128
Learners showed evidence of knowing how to prepare a
a) Definition of bar graph
Line 2a: The learners prepared a frequency table as displayed in table 4.5.4bi.
frequency table as they had been taught it previously (line 2a).
Line 2b: Teacher D defined and described a bar graph orally and wrote it on the chalkboard: ‘It is a statistical graph
used in representing data in the form of a bar. A bar graph is used for representing discrete data. When a bar graph is
used to represent information, you can easily see the information physically and understand how one discrete piece of
Teacher content knowledge was used to define and explain bar
graph construction and its uses (line 2b).
data is different from another. A bar graph can be represented vertically or horizontally.’ The next step was for the
teacher to demonstrate how a bar graph is constructed..
Instructional skills such as construction skill involving the
Construction of bar graph
drawing of the axes, choosing of scale, labelling of axes,
Line 2c: Teacher D described this on the chalkboard as follows: ‘You draw vertical and horizontal axes and
plotting of points, and joining the line of best fit were utilised in
labelled them (the horizontal axis represents the frequencies, and the vertica axis represents the companies). The scale
constructing a bar graph (line 2c).
of the horizontal axis were determine by considering the lowest and the highest value of the number of cars and
appropriately labelling the horizontal axis with names of the companies. In learners’ mother tongue, he said, labella ga
ke go bontsha, meaning ‘Watch me as I demonstrate it’. Teacher D continued, ‘ For the first bar Tata, the frequency is
Teacher D taught a bar graph using a procedural knowledge
3; For the second bar, the frequency is 10; for the third bar, the frequency is 8 etc.’
approach (line 2c) (content knowledge and instructional
strategy).
Graph construction skills of drawing the axes, choosing scale,
labelling axes, plotting points, and joining the line of best fit
were used to construct a bar graph (line 2c).
The learner’s mother tongue was used to direct the learners’
attention to the lesson and reinforce their comprehension of the
material (line 4b) (instructional strategy) (line 2c).
129
Figure 4.5.4a:
Bar graph showing the numbers of makes of cars in a car park
Teacher content knowledge was made use of to interpret the
a) Interpretation of bar graph
Line 3: Teacher D drew the bar graph, as in Figure 4.5.4a, and interpreted it by indicating that Tata was the least
bar graph (line 3).
frequent make of car in the car park, while BMW was the most frequent. The second most frequent was Toyota.
Line 4a: Teacher D asked, ‘Why do you think the most frequent make of car in the car park was BMW?’ Learners
Oral questioning (instructional strategy) was used to probe
answered one by one and gave the following answers: ‘BMW produce the most popular cars.’ ’BMW produce
learners’ views about the most frequent make of car (line 4a).
prestigious cars,’ ’BMW produce cars of high quality,’ etc.
Line 4b: Teacher D further answered the question, ‘BMW produced the highest number of cars in the car park.’ In
their mother tongue he said, ‘ke mang a sahlaloganyeng, meaning ‘who does not understand the explanation?’
Open-ended questions that called for reasoning and
analytical skills (line 4a). Reasoning skills were employed to
arouse interest and focus the learners’ minds on the construction
and interpretation of the bar graph.
130
Instructional strategy was to set an activity on bar graph
b) Classwork
Line 5a: Teacher D set the learners an activity to solve individually. It involved a table of values of the distribution of
construction which learners had to solve individually (line 5a)
marks obtained by 50 learners in a class test. Learners were asked to construct the bar graph and calculate the
percentage of learners who failed the test if the pass mark was 5 out of 10 or 50%.
Learners solve activity of bar graph individually as a way of
Table 4.5.4bii:
Frequency table showing the mark distribution of learners in a class test
assessing how well they have understood what the teacher
taught them (line 5b).
Marks
1
2
3
4
5
6
7
8
9
10
Frequency
3
1
2
7
10
12
9
3
2
1
Line 5b: Learners solved the question individually by constructing the bar graph and determining the percentage of
learners that had failed the test.
Line 6: While the teacher monitored how the learners were progressing in their classwork, he offered additional
Learners asked the teacher to explain why there should be
explanations for labelling the axes and drawing the bars. For instance, one of the learners asked, ‘Why do you leave
constant spaces between the bars, meaning that they did not
equal spaces between the bars when the companies produce a different number of cars in the car park?’
understand this from the earlier explanation that the teacher had
provided using procedural knowledge (line 6). Learning
difficulty of their lack of understanding of the construction of
bar graph was discovered by Teacher D.
Line 7a: Teacher D explained: ‘All the companies manufacture cars only, but of different makes, hence they have to be
Teacher content knowledge was used to explain why there
separated by equal spacing by choosing appropriate scale, which differentiates one make of car from another. The
should be constant spacing between the bars (line 7a).
difference in height of the bars is because of the difference in the number of cars produced. In terms of your classwork,
the differences in the height of the bars are as a result of the number of students which correspond to the marks they
scored,’ the teacher said. A conceptual knowledge was used to explain the frequencies, the cars manufactured and
while there should be constant spacing between the bars.
Teacher used conceptual knowledge requiring the drawing of
the bars with constant spacing based on the company and the
scale that was chosen for constructing the graph and the
differences in height resulting from the varying frequencies of
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Line 7b: Learners showed evidence of a grasp of the lesson as they constructed the bar graph more efficiently,
especially after the teacher demonstrated how to construct the bar graph using a conceptual knowledge approach.
the cars manufactured by each company (line 7a).
Learners demonstrated evidence of a grasp of the lesson as they
constructed the graph more efficiently (line 7b).
Line 8: Teacher D identified learners’ difficulty in constructing bar graphs during the monitoring of the learners while
Teacher D identified misconceptions, involving drawing a
they are doing classwork, such as unequal spacing between the bars (as most learners think this merely indicates the
histogram instead of a bar graph through not considering the
space and bars without considering the sizes),. For example, while most learners used the space between the first bar
spacing between the bars, during the examination of their work
and the horizontal axis to determine the spaces between the other bars, some did not consider the consistency of the
on bar graph construction (line 8).
spacing between the bars, irrespective of the size of the space between the first bar and the horizontal axis, as in Figure
4.5.4a. Some learners drew the bar graph with different spacing between the bars, and others drew histograms instead
of bar graphs.
Another misconception concerns the inconsistency in spacing
and sizes of the bars (line 8).
Line 9: These misconceptions (as stated above) in which learners drew a histogram instead of a bar graph and drew the
Extra elucidation on how to construct and interpret a bar graph,
bars without considering the size of the latter were addressed by Teacher D through extra explanations to individual
especially with respect to the drawing of the histogram instead
learners as well as by compulsory additional activities from the textbook which the learners did in class individually.
of a bar graph and inconsistency of spacing between the bars,
was offered on a one-on-one basis to correct the misconceptions
and learning difficulties (line 9). Extra class activities was given
to the learners’ to deepen their understanding of bar graph
construction (line 9).
Line 10a: During the lesson, Teacher D repeated what he said in line 2b and 2c and provided further explanations on
Conceptual knowledge was used to explain the meaning of a
the meaning of a bar graph, construction of bar graph with emphasis on the space between the bars drawn according to
bar graph and how it can be constructed by considering the
scale, the size of the bars and the consistency of the space between the bars individually to some learners who were
frequency and drawing the bars with appropriate scale. How the
experiencing difficulties. For example, one of the learners whose classwork had been marked wrong, because she had
scaling affected the consistency of the spaces between the bars
constructed a histogram instead of a bar graph, requested clarity as to why her answer was wrong.
and sizes of the bars, and the learners’ misconceptions and
learning difficulties (inconsistency of spaces between the bars
Line 10b: Teacher D stated that the learner had not left spaces between the bars, as explained in the example on the
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and sizes of the bars) (Line 10b) were addressed. Teacher used
chalkboard. The spaces between the bars in a bar graph help to differentiate between categories of data (companies)
content knowledge and instructional strategy to explain
and must be equal because we are dealing with cars, though of different makes (categories). ‘In a bar graph, there
conceptually the construction and interpretation of bar graph
should be a constant spacing between the bars and the sizes of the bars must be the same,’ he said.
(line 10b) to the learners.
Line 10c: The learner nodded her head in agreement with the teacher’s explanation, as explained in line 10b. Teacher
D corrected the classwork, and the learners wrote down the corrections in their class workbook.
Teacher content knowledge was used to address learners’
misconceptions and learning difficulties using teacher’s
conceptual knowledge (line 10b)
Line 11: At the end of the lesson, the learners were given homework from the school supplementary textbook,
A supplementary mathematics textbook was used as a source
of information for teaching bar graph and assigning homework
(line 11).
CLASSROOM PRACTICE (SECOND LESSON OBSERVATION)
CATEGORISATION/THEMES
Topic: Construction, analysis, and interpretation of histograms. Class: Grade 11
Line 1: After greeting the class, Teacher D began the lesson on histogram construction by checking and marking
Knowledge of stem and leaf diagrams is regarded by the teacher
homework on the construction and interpretation of stem-and-leaf diagrams. The teacher and learners provided
as an important part of learner’s prior knowledge before the
corrections to the homework so that learners who experienced difficulties could correct their mistakes. While providing
histogram can be successfully taught to learners, Checking
the corrections, Teacher D explained once more how a stem-and-leaf diagram is constructed by arranging the leaves in
learners’ homework on the construction and interpretation of
the right-hand column and the stem in the left-hand column. ‘Just as the stem-and-leaf diagram is used to represent
stem-and-leaf diagrams was used as an instructional strategy
group data, the histogram we are about to study now is also used to represent grouped data,’ he added.
to introduce the lesson and to determine learners’ background
knowledge or conceptions in histogram construction (line 1)
(teacher’s PCK).
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Line 2: Learners did corrections, which were written on one side of the chalkboard by Teacher D, while he wrote the
Teacher C wrote the new topic while learners corrected their
new topic on the other side of the chalkboard.
mistakes in their homework (line 2).
Line 3: Teacher D presented a photocopy of an activity on the construction of a histogram representing the mass of
Instructional strategy of using photocopied material to provide
each player in a 2003 South African rugby squad. The masses of the 30 players were: 115, 122, 110, 110, 105, 112, 80,
a source of information for lesson activity was used to set
98, 90, 93, 85, 87, 99, 84, 112, 76, 96, 128, 110, 108, 118, 105, 108, 118, 90, 89, 90, 88, 103, and 85 kg. The activity
exemplar questions to demonstrate the construction and
requests the learners to: a) prepare a frequency table of the data presented with a class of 10; b) use the frequency table
interpretation of histogram (line 3).
to construct a histogram; and c) determine from the histogram (i) the mean; (ii) interval that has the highest frequency;
(iii) percentage of players whose weight fell between 110 and 120 kg and (iv) the mode.
Teacher D instructed learners to prepare a frequency table from
a) Preparation of frequency table
Line 4: Learners were instructed to prepare a frequency table by calculating the frequencies of each interval. The class
the raw data presented (line 4) (Instructional strategy).
boundaries, mid-values, and fx were later calculated to help in answering question (b) and (c), as normally done if the
need arises, or based on the questions in the learners’ activities (see table 4.5.4c), and to calculate the measures of
central tendency (the mean, and the mode) that best describe the masses of the players. The instruction presupposed
that learners knew how to prepare a frequency table; hence class boundaries, mid-value and fx were not explained.
Line 5a: The frequency table was constructed by the teacher and the learners. While Teacher D wrote down the
Teacher content knowledge on the preparation of frequency
frequencies, learners counted the masses within each interval. The mid-values were calculated by finding the average
tables was used to create a frequency table, and to explain how
of the upper and lower class of each class interval while fx was calculated by finding the product of mid-value (x) and
to prepare the frequency table of grouped data by grouping the
frequencies (f) of the individual classes, row by row.
data according to class; also to determine the frequency as well
the class boundaries, mid-values and fx and calculating
measures of central tendency, as indicated in questions (b) and
(c) (line 5a).
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Table 4.5.4c: Frequency table showing the masses of players in the 2003 South African rugby squad
Teacher procedural knowledge was used in preparing the
frequency table with learners (line 5a).
Class intervals
Class boundaries
Freq. (f)
Mid-values (x)
Fx
70-79
70-80
1
75
75
80-89
80-90
6
85
510
90-99
90-100
7
95
665
100-109
100-110
5
105
525
110-119
110-120
9
115
1035
120-129
120-130
2
125
250
Procedural knowledge was utilised to describe how a
histogram should be constructed, an approach that the teacher
∑f = 30
∑fx = 3060
Line 5b: Teacher D defined and described a histogram orally and wrote it down on the chalkboard as indicated in the
textbook: ‘A histogram is a graphical representation, showing a visual impression of the distribution of grouped data.
It consists of tabular frequencies shown as adjacent rectangular bars, erected over discrete intervals, with an area
equal to the frequency of the observations in the interval. Unlike the bar graph, a histogram is used to represent a
large set of data (e.g. a population census) visually, but with no spaces between the bars,’ the teacher said. After the
explanation, he referred to the frequency table and indicated the usefulness of the table in the construction of the
histogram beginning with the class boundaries, followed by the frequencies. He thereafter began to demonstrate how to
construct the histogram.
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felt would make the histogram more accessible to the learners
(teacher topic-specific content knowledge and instructional
strategy) (line 5c).
Teacher content knowledge was used to explain the usefulness
of the frequency table in constructing a histogram, beginning
with the class boundaries, and followed by the frequencies (line
5b).
Topic-specific construction skills of drawing the axes,
choosing scale, labelling axes, plotting the points and joining
the line of best fit were used to construct a histogram
(instructional skill) (line 5c).
a) Construction of a histogram
Line 5c: Teacher D illustrated the histogram construction visually using procedural knowledge by drawing the vertical
and horizontal axes and labelling them using a scale chosen by the teacher by considering the lowest and highest
values of the frequencies, with the vertical axis representing the frequencies, and the horizontal axis representing the
masses on the chalkboard. He drew two bars of the histogram and instructed learners to complete it according to the
class boundaries and frequencies.
Line 6a: Learners completed the histogram individually in their workbooks while Teacher D monitored and examined
their responses. Most of the learners who had correctly completed the table drew a histogram, as shown in Figure
4.5.4b. Other learners who had not drawn their histogram correctly because of incorrect scaling and labelling of the
horizontal axis, among other errors, and also because of lack of comprehension, corrected their mistakes by copying
the correct diagram presented on the chalkboard. Some learners drew bar graphs instead of histograms by leaving
Learning difficulties of drawing a bar graph instead of a
histogram were identified through analysis of learners’
responses to classwork (line 6a).
spaces between the bars. The difficulties experienced in scaling could have arisen because at the beginning of the
Insufficient teacher explanation (pedagogical knowledge) of
activity the teacher did not describe and explain how to choose a scale for constructing a graph of grouped data.
choosing the scale for constructing a histogram with a
procedural approach led to learners constructing a bar graph
instead of histogram (line 6a).
Figure 4.5.4b:
Histogram showing the distribution of the masses of players in a 2003 South African rugby
squad
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Line 6b: After completing the histogram, Teacher D answered follow-up questions (see line 3) such as question (c)
Learners drew a bar graph instead of a histogram (line 6a)
which requested the learners to determine (i) the mean; (ii) the interval that had the highest frequency; (iii) the
(misconception).
percentage of players whose weight fell between 110 kg and 120 kg and (iv) the mode from the histogram.
c) (i) Calculating the mean
The mean was calculated: ‘Mean =
Teacher’s procedural knowledge was used in calculating the
∑
∑f
fx
mean (line 6b). This was done by substituting the values in the
=
3060
= 102 kg’; the teacher said
30
equation:
∑ fx
∑f
=
3060
30
= 102kg
ii) Identifying the interval with highest frequency
Teacher content knowledge was used to analyse and interpret
Line 7a: Teacher D analysed the histogram (in which learners determined which interval (110–119) kg had the highest
the histogram (line 7a), demonstrating the application of
frequency, and which intervals had the next highest frequencies (90–100) kg). He then wrote the answer, ‘The class
analytical and interpretational skills by calculating the class
with the highest frequency is 110–119 kg’. ‘Any question about how we determine the class that has the highest
with the highest frequency.
frequency?’ he asked. As there was no question from the learners, he answered the next question about the percentage
of learners that fail the test.
Procedural knowledge was made use of to demonstrate how to
iii) The % of players that fall within (110–120) kg =
9 100
x
= 30%
30
1
determine mode from a histogram (line7b) (instructional
strategy).
From Figure 4.5.4b, it was determined that the individual mass of most of the players (9 out of 30) in the squad fell
between 110 kg and 120 kg, which formed 30% of the players in the squad.
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iv)
Calculating mode from a histogram
Line 7b: Teacher D then determined the mode using procedural knowledge by drawing a diagonal line from the topright corner of the highest bar to the top-right corner of the bar next to it on the left-hand side, and drawing a diagonal
line from the top-left corner of the highest bar to the top-left corner of the bar next to it on the right-hand side (as in
case A). A line was drawn from the meeting point of the two diagonals down to the horizontal axis to locate the mode.
‘After the identification of the interval where the mode will be located, the diagonal lines help to locate the mode
Teacher content knowledge and instructional knowledge
were employed to demonstrate how to determine the mode from
the histogram by drawing intersecting diagonals and using the
point of intersection to locate the mode (line 7b).
within the class interval,’ the teacher said. ‘By drawing a line from the point of intersection of the diagonals, the mode
was located as 113kg (see Figure 4.5.4b),’ he added.
Line 8: After rule-oriented procedural knowledge was used to demonstrate how to calculate the mode from the
Learners wrote down in their notebooks what the teacher had
histogram, learners were given time to write the explanation of how the mode was calculated from the histogram that
explained as he instructed them.
Teacher D had written on the chalkboard into their workbooks. “Now you can write down the explanation I have given
on the chalkboard into your workbooks,” the teacher said.
Teacher’s instructional knowledge was used to provide time
for the learners to write down the explanation given by him on
how to calculate the mode.
Line 9: Classwork based on construction and interpretation of bar graphwas then given to the learners to solve
A supplementary recommended mathematics textbook was
individually from their supplementary textbook. Learners had to complete their classwork in their workbooks at home,
employed as a source of information for teaching histograms
as they were not able to complete it by the end of the lesson period.
(line 9).
Using a classwork (line 9) assignment for feedback was part of
the teacher’s instructional strategy during the lesson.
Line 10: When the lesson was about to end and learners were still busy doing the classwork; a learner enquired
A misconception was identified through oral questioning from
(referring to Figure 4.5.4b), ‘why it was necessary to label the horizontal axis from 70, and not from 0, as was done on
the learners on the labelling of the data axis (line 10).
the vertical axis?’ This question demanded a conceptual knowledge approach, which was provided in line 11.
Line 11: Teacher D replied that, ‘One labels the horizontal axis from 70, because 70 is the lowest value on the table.
Teacher’s conceptual knowledge was used to clarify the
In addition, a scale of 1cm = 10 units was used to label the data axis. Therefore, if you begin from 0, all the values as
reason that it was necessary to start labelling the horizontal axes
indicated on the table of values will not be accommodated on the graph paper provided,’ he added. Alternatively, ‘One
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can label from 0 and make a continuation line between the 0 and 70. The continuation line indicates that the intervals
below 70 have been omitted so that the graph can be contained on the graph paper,’ the teacher said. A related
example was drawn from the same supplementary mathematics textbook.
from 70 (line 11).
Teacher content knowledge and instructional strategy were
applied to explain conceptually why it was not necessary to start
labelling from zero as a result of the scale of 1cm = 10 units,
which was chosen because of the dimensions of the graph paper
(line 11).
Line 12: More learners seemed to be satisfied with the teacher’s explanation by using a conceptual knowledge
While some learners indicated that they were satisfied with the
approach as explained in line 11. They nodded their heads, while a few others were still experiencing difficulties and
teachers’ explanation, others felt that the teacher had not cleared
shook their heads which may be as a result of lack of understanding due to inadequate explanation regarding why the
up the difficulty (line 12).
labelling of the data axis has to start with 70 and not 0 .
Insufficient teacher content knowledge was made use of to
address learners’ difficulties in labelling the data axis correctly
(line 12).
Line 13: Teacher D gave them homework and promised to organise extra tutoring after normal school hours, where he
The instructional strategy of employing homework (line 13)
would try to explain once more how to construct, analyse, and interpret a histogram using activities related to everyday
to assess how well learners understood the lesson was adopted
life.
during the lesson. Extra tutoring was also proposed for helping
learners with difficulties.
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Summary of lesson observation of Teacher D
Teacher D demonstrated aspects of procedural knowledge of the topics of bar graph and
histogram construction. He combined appropriate pedagogical knowledge of teaching bar
graphs and histograms with a rule-oriented procedural and conceptual knowledge approach.
The content knowledge of bar graph and histogram construction used for teaching the
observed lessons was both procedural and conceptual, but mostly procedural. For example,
Teacher D demonstrated procedurally how bar graphs and histograms are constructed using
the construction skills of drawing the axes, and choosing a scale by considering the lowest
and highest values of the data and frequencies as well as the dimension of the graph paper
provided. The next step was to plot the points and draw the line of best fit (ref Section 5.5.4,
first lesson observation and line 2c; second lesson observation, and line 5c). In terms of his
conceptual knowledge, he explained how histograms should be constructed with a scale, even
when data values do not start from zero, so that the values can be accommodated on the graph
paper provided (refSection 4.5.4, second lesson observation, and line 11) when he discovered
that the learners were experiencing some difficulties.
At the beginning of the lesson, Teacher D used his pedagogical knowledge of instructional
skills and strategies to try to identify learners’ preconceptions by giving them a pre-activity
on the preparation of a frequency table, and by checking and marking their homework on
stem-and-leaf diagrams. Through the pre-activity, learners demonstrated that they had
mastered the concept of preparing a frequency table of ungrouped data and of constructing
bar graphs because they had been taught these in the past (ref Section 4.5.4, first lesson
observation line 1). But checking and marking learners’ homework on stem-and-leaf
diagrams revealed that some learners had experienced difficulties that could have been the
results of inadequate explanation or of lack of comprehension by the learners (ref Section
4.5.4, second lesson observation, and line 1). These difficulties were corrected before the new
lesson began. In the lesson observed, Teacher D knows that stem-and-leaf diagrams are
necessary for histogram construction. There is no evidence in his lessons that he knows of the
misconceptions his students are likely to have of bar graph and histogram construction.
Hence, he can be said to have provided poor and inadequate explanations that resulted in
certain learning difficulties. This is possibly understandable because the topic of data
handling is a new one. Learners’ misconceptions and learning difficulties were identified
140
through marking and analysing the learners’ classwork, as well as through oral questioning,
where learners could request clarification of what they did not understand about determining
the mode from a histogram. These misconceptions and learning difficulties were not
adequately addressed through individual problem-solving class activities and further
explanations on the construction and interpretation of bar graphs and histograms, because
some learners continued to experience difficulties. For example, when the lesson was about
to end and learners were doing the classwork, a learner enquired (referring to Figure 4.5.4b),
‘Why is it necessary to label the horizontal axis from 70, and not from 0, as was done on the
vertical axis?’ (ref Section 4.5.4, second lesson observation, and line 10). Teacher D replied
that, ‘One labels the horizontal axis from 70, because 70 is the lowest value on the table. In
addition, a scale of 1cm = 10 units was used to label the data axis. Therefore, if you begin
from 0, all the values as indicated on the table of values will not be accommodated on the
graph paper provided,’ he added. Alternatively, ‘One can label from 0 and make a
continuation line between the 0 and 70. The continuation line indicates that the intervals
below 70 have been omitted so that the graph can be contained on the graph paper,’ the
teacher said. A few learners shook their heads to indicate that they had not understood the
explanation.
Teacher D probably does not command sufficient content and pedagogical
knowledge to address learners’ misconceptions and learning difficulties effectively in this
respect.
4.6
Video recordings of lesson observation of the four teachers
The video recordings of the four participating teachers confirmed the teaching of the
construction and interpretation of bar graphs, histograms, ogives, box-and-whisker plots, and
scatter plots during lesson observations (see Section 4.5.1–4.5.4). The video recordings were
also used to triangulate the written notes taken during classroom observations.
4.7
4.7.1
Teacher development of PCK
Teacher development of subject matter content knowledge
In the interviews, the teachers claimed that they had studied mathematics and general method
courses at university, which helped them to adapt the way they taught school statistics (ref
Appendix XVII, items 1, 2 and 3) by employing appropriate instructional skills and strategies
to teach statistical graphs. For instance, when they were asked, “If one of the courses you
studied at university is mathematics methodology, how did it help you to prepare for your
141
lessons for teaching?” Teacher A indicated that the method course he had studied helped him
to vary his instructional strategies (ref Appendix XVII, item 5a). Teacher B asserted, “The
mathematics method courses help me to vary formulae and strategies for teaching statistics.”
Teacher C averred that the courses had helped him to prepare his lessons in line with the
objectives of the lessons. And Teacher D said the courses helped him to plan his lessons in
line with the work schedules, assessment and evaluation of his lessons.
The participating teachers were further requested to indicate how they knew that their
teaching in statistics was effective, as a way of establishing whether the contents of statistics
lessons are adequately delivered by teachers with content knowledge of statistics. Teacher A
claimed that through analysis of the learners’ responses to classwork, homework, and
assignments, he knew that his lessons were effective (ref Appendix XVII, item 8). Teachers
B, C and D said virtually the same thing, which means that the teachers may have
demonstrated the content knowledge of school statistics which they possess during their
lessons.
To further ascertain how the participating teachers gained their content knowledge for
teaching, they were asked, “Have you attended a mathematics workshop or teacher
development programme?” and also, "as a mathematics teacher, did you benefit from the
workshop?” Teachers A, B and C responded that they had attended workshops on data
handling (the new topic in the curriculum) and learnt how to teach challenging topics in this
respect. Teacher D responded: “Yes, I attended many workshops on teacher development in
content knowledge especially in data handling. I did not benefit much because I was taught
what I already know in mathematics”, which could mean that Teacher D became more aware
that he already possessed the required content knowledge for the subject he was teaching.
From the above analysis, the teachers can be said to have developed their content knowledge
in statistics teaching through formal education, which gave them the opportunity to study
mathematics and the methodology of teaching and enabled them to design instructional
strategies for carrying out effective teaching. Through classroom practice, lesson planning
and preparation, and content knowledge workshops, they gained further content knowledge.
The teacher portfolios and concept mapping exercise confirmed that the teachers possess the
content knowledge of school statistics as they listed the subject matter content of school
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statistics to be taught in a sequential and logical manner (ref Appendix XXI, teachers’
portfolios; Section 4.4).
In addition to listing the content of school statistics, the participating teachers taught
statistical graphs using both procedural and conceptual knowledge approaches following the
learning outcome of data handling as stipulated in curriculum (DoBE, 2012) and how
graphing concepts should be taught (Flockton et al, 2004; Leinhardt et al, 1990) in their
lessons on statistical graphs. Using topic-specific content knowledge and instructional skill
(construction skill) of drawing the axes, choosing of scale, labelling of the axes, plotting the
points and joining of the line of best fit, Teacher A for instance, demonstrated procedurally
how to construct a histogram (ref Section 4.5.1, first lesson observation, and line 9). While
some learners displayed evidence of grasp of their lesson, a few experienced some learning
difficulties (ref Section 4.5.1, first lesson observation, and line 11) which resulted in the
teacher adopting a conceptual knowledge approach to assist learners who are experiencing
some difficulties (ref Section 4.5.1, first lesson observation, line 16). Thus, the participating
teachers can be said to have mastered the content of school statistics which they developed
through formal education and classroom practice, and demonstrated it by teaching with
procedural and conceptual knowledge approaches, using recommended textbooks, a work
schedule and by attending content-driven knowledge workshops.
4.7.2
Teacher development of pedagogical knowledge (instructional skills and
strategies)
The focus of this section was to determine the instructional skills and strategies that the
participating teachers utilised in teaching school statistics. The teacher questionnaire, lesson
observation, written reports and documents analysis were used to collect data to ascertain the
teachers’ pedagogical knowledge in statistics teaching. The purpose of the questionnaire was
to establish what the teachers actually did while teaching assigned topics in school statistics
and to determine the pedagogical knowledge (instructional skills and strategies) they possess
and use in teaching school statistics.
In their responses to the questionnaire (ref Appendix XVIII), the teachers claimed they had
achieved the objectives of their lessons, in which learners are expected to construct, analyse
and interpret statistical graphs, and apply the knowledge to everyday real life situations
according to the learning outcomes of data handling (DoBE, 2010). This means that the
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teachers applied content and pedagogical knowledge that was adequate to elicit
understanding of school statistics. For example, they were asked, “Do you think that the
learners achieved the objective of the lesson and if not, what do you do to improve their
understanding?” to establish what strategies they adopted and how good these strategies were
(ref Appendix XVIII, item 7). All four teachers claimed they knew that the objectives of their
lessons had been achieved through active participation of learners in their lessons, and
responses to classwork, homework, assignments, tests, and examinations in statistics (ref
Appendix XVIII, item 7). Teacher A tried to engage the learners in extensive class
discussions to improve their understanding of statistical graphs, while Teacher B used
teaching aids such as statistical charts and an overhead projector to display statistical
diagrams. Teacher C indicated that he made use of extra class activities related to real life to
improve learners’ understanding of the lessons, whereas Teacher D claimed that he used
additional examples and past questions in tests to improve learners’ understanding of
statistical graphs.
From the responses of the four teachers to the questionnaire, it can be understood that they
gained their pedagogical knowledge through classroom practice, which involved planning
and presentation of lessons, as well as using classwork, homework, exams and assignments,
to assess how well learners understood the lessons on statistical graphs. The participating
teachers taught statistical graphs with instructional strategies which they felt could help
learners to understand the topics and learners responded positively to classwork, homework
and assignments. They also claimed to have used class activities related to familiar real life
and problem solving on past test questions in statistics to help learners improve their
understanding of statistical graphs. The lesson observation, teacher written reports, and
document analysis confirmed that the teachers used class activities related to familiar real life
situations, problem solving in the form of drill and practice, as well as employing classwork,
homework and assignments to assess how well learners had understood the lessons on
statistical graphs. For example, during the lesson observation on scatter plot construction,
Teacher C made use of the age distribution of persons infected with HIV/AIDS in two towns
(familiar real life situation) as classwork to assess how well the learners understood his lesson
on the construction and interpretation of scatter plot (ref Section 4.5.3, second lesson
observations, and line 9). The teachers also utilised both procedural and conceptual
knowledge approaches in teaching statistical graphs (ref Section 4.5.4, first lesson
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observation, line 2c and 7a). In the teacher’s written report, Teacher D indicated that he
tackled learners’ learning difficulties by adopting different teaching approaches and
providing additional class activities related to real life (ref Appendix XX, item 6).
In the learners’ notebooks (ref Appendix XXI, learner workbooks) there are examples of
statistical graphs, calculations and exercises related to the concepts they were taught
according to the procedures for constructing statistical graphs, indicating also that the
teachers may have used a procedural knowledge approach. For example, the workbooks of
learners in Teacher A’s class displayed diagrams of histograms constructed as examples by
the teacher and others done as classwork by drawing the axes, labelling the axes based on a
given scale, plotting points, and drawing lines of best fit (ref Appendix XXI, learner
workbooks). Teachers B, C and D’s learner workbooks (ref Appendix XXI, learner
workbooks) contained similar records of examples in which a procedural knowledge
approach may have been used for teaching statistical graphs. The conceptual knowledge was
used less frequently to assist learners that were experiencing some learning difficulties (ref
Section 4.5.3, second lesson observation, and line 4dii). All four teachers made use of
classwork, homework and assignments as well as the SBA to assess how well learners
understood the lessons on statistical graphs. The assessment tasks appeared to be similar
because the four participating teachers used the same assessment guidelines, work schedules
and textbooks as recommended by the Department of Basic Education (ref Appendix XXI,
teacher and learners’ portfolios) for teaching Grade 11 mathematics. Learners’ recorded
examples from extra lessons (ref Appendix XXI, learner workbooks) indicating that the
teachers must have individually conducted extra tutoring to help learners who experience
learning difficulties (inability to choose scale of grouped data) in order to deepen their
understanding of data handling.
From the above discussion, it is evident that the participating teachers used predominantly a
procedural knowledge approach and to some extent a conceptual knowledge approach,
construction skills, extra tutoring, examples drawn from familiar real life situation, additional
class exercises in the form of drill and practice in the teaching of statistical graphs. By doing
so, the teachers may have developed more knowledge of the instructional skills and strategies
for teaching school statistics.
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4.7.3
Teacher development of knowledge of learners’ preconception and
learning difficulties
A teacher questionnaire, lesson observation, written reports and documents analysis were
used to investigate whether the teachers had knowledge of learners’ preconceptions and
misconceptions, if any, as well as of learning difficulties about statistical graphs such as bar
graphs, histograms, ogives, and scatter plots. The investigation revealed that despite many
years of teaching experience held by the participating teachers, they possessed no knowledge
of learners’ preconceptions in statistical graphs. For instance, in the questionnaire, they were
asked, “What prior knowledge does your lesson require?” Teachers A and D claimed that
learners need measures of central tendency as prior knowledge for bar graphs, histograms and
ogives construction (ref Appendix XIX, item 4). Teacher B said that learners need simple
addition and subtraction skills, as well as measures of central tendency as prior knowledge
for bar graph and ogive construction. Teacher C asserted that learners need to understand
measures of central tendency and know how to interpret information from straight-line graphs
as prior knowledge for scatter plot and ogive construction. All their responses indicated that
they had acquired previous knowledge about the topics they were teaching. But what was
needed was the knowledge the learners had before they were taught the concept of statistical
graph (preconception). It means that the instructional strategies adopted by the teachers could
not elicit learners’ preconceptions of the various topics they taught depicting the fact that the
teachers have no knowledge of learners’ preconceptions in statistics teaching.
The teachers were also asked, “How did you identify the prior knowledge (preconceptions)
about statistical graphs with which the learners came to the class?” Teachers A and C claimed
that they used probing questioning to establish if learners had gained prior knowledge of
measures of central tendency linked to histograms, ogives and scatter plot construction (ref
Appendix XIX, item 4–6). This was confirmed in the lesson observation of Teacher A (ref
Section 4.5.1, of the first lesson observation, and line 1) in which learners mentioned mode,
median and mean when the teacher attempted to probe their preconceptions of histogram
construction. Teacher B claimed that he determined their prior knowledge in statistical graphs
constructions while correcting their responses to homework and using pre-activities related to
the topic he was going to teach (ref Appendix XIX, item 6). This was confirmed in the
observation of a bar graph construction lesson given by Teacher B (ref Section 4.5.2), of the
first lesson observation, and line 1) in which learners used knowledge of simple addition to
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prepare a frequency table in a pre-activity in ungrouped data. Learners also mentioned
different ways of representing data as prior knowledge for ogive construction. Teacher D
claimed that he made use of pre-activities and probing questions to determine prior
knowledge in statistical graph constructions such as bar graphs and histograms (ref Appendix
XIX, item 6).
This employment of pre-activities and oral probing questions was confirmed in the lesson
observation of Teacher D (ref Section 4.5.4, of the first lesson observation, and line 1) who
used pre-activities, and checking and marking learners’ homework, to attempt to identify
learners’ preconceptions of bar graphs and histogram construction.
From the responses of the participating teachers to the questionnaire, it appears that they
have used topic-specific instructional strategies such as asking oral probing questions,
checking and marking learners’ homework, and utilising pre-activities at the beginning of the
lessons to try to identify learners’ prior knowledge in the topics taught in statistical graphs.
By employing these strategies, all four teachers could have been adjudged to have
demonstrated that they knew about the learners’ possible preconceptions and were therefore
able to decide which instructional strategy was best to elicit the prior knowledge that was
essential for the learning of the new concepts. But the strategies only elicited learners’
previous knowledge and not the preconceptions, which means the teachers possess no
knowledge of the learners’ preconceptions. The teachers’ written reports and documents
analysis confirmed that the participating teachers tried to identify learners’ prior knowledge
in statistical graphs using diagnostics techniques such as oral probing questioning, preactivities as well as checking and marking of learners’ homework (ref Appendix XIX, items 8
and 9; Appendix XXI, teacher portfolios).
Regarding the learners’ misconceptions and learning difficulties, all the participating teachers
adopted monitoring and analysis of learners’ responses to classwork to identify any
misconception and learning difficulty that the latter may experience during their lessons on
statistical graphs. As noted in their responses to the interview (ref Appendix XX, item 14),
the learners’ learning difficulties range from basic computations of mode, median and mean
of grouped data (as in the case of teacher A), to choosing of the scale for constructing graphs
of grouped data (for Teachers B and C), and determining the mid-points of graphs of grouped
data. From the teachers’ responses to the questionnaire, while Teachers A and C addressed
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the difficulties by giving learners additional exercises in graphs of grouped data, Teacher B
did so by specifically teaching the learners how to choose different scales for different data
for the sake of uniformity in graph construction. Teacher B tackled the learners’ difficulties
in graphs of grouped data by giving them additional examples and possibly repeating the
lesson in order to reinforce learners’ understanding of statistical graphs. The teachers were
further asked, “What is it about statistics that makes it easy or difficult?” Teachers A and B
said that measures of central tendency represent an easy concept to learn. Teacher C
commented that relating statistics to real life makes it lively, interesting, and easy to learn.
Teacher D said that statistics is easy to learn if someone who is knowledgeable presents the
topic. Therefore, teacher content knowledge of a topic should be adequate in order to make
the teaching of statistics comprehensible and accessible to the learners.
In the document analysis, misconceptions such as drawing a histogram instead of a bar graph,
as in the case of Teacher B, and drawing a bar graph instead of histogram, as in the cases of
Teacher A, C and D, were addressed individually through extra tutoring, extra class activities
and post-teaching discussions in statistical graphs (ref Appendix XX1, teacher portfolios)
during and after school hours.
The lesson observations and the teacher written reports confirmed that the teachers identified
learners’ misconceptions and learning difficulties by monitoring and analysis of learners’
responses to classwork, homework and assignments in statistical graphs and addressing the
misconceptions and learning difficulties by extra tutoring, teaching learners how to choose
scale, re-demonstrating or repeating the lessons, extra class activities and post-teaching
discussions in statistical graphs. For example, the learners’ misconception of drawing a
histogram instead of an ogive (ref Section 4.5.2, second lesson observation, and line 7a) and
the learning difficulty emanating from the misconceptions of interpreting a negatively
correlated scatter plot as having no correlation due to an outlier (ref Section 4.5.3, second
lesson observation, and line 4bii) were identified during the monitoring and analysis of
learners’ responses to classwork by Teachers B and C on ogive and scatter plots respectively
(ref Appendix XX, items 1 and 2). The misconceptions and learning difficulties were
addressed by post-teaching discussion (ref Section 4.5.3, second lesson observation, and line
12) and extra class activities in the form of drill and practice (ref 4.5.2, first lesson
observation, and line 15).
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From the above analysis, it can be concluded that the individual participating teachers
developed their knowledge of learning difficulties through analysing and monitoring learners’
responses to classwork, homework and assignments to identify learners’ learning difficulties
in statistical graphs. The teachers also extended their knowledge of these difficulties by
addressing the difficulties using additional tutoring, extra class activities, post-teaching
discussions, re-teaching, and further explanation of the lessons they taught, individually to
learners during and after the lessons.
4.7.4
Teacher development of PCK in statistics teaching
By summing the ways through which the participating teachers developed the subject matter
content knowledge, pedagogical knowledge and knowledge of learners’ preconceptions and
learning difficulties, one would be able to determine how the participating teachers developed
their PCK in statistics teaching. In section 4.7.1, it was deduced that the participating teachers
possess the content of school statistics which they acquired through formal education, and
demonstrated it by employing procedural and conceptual knowledge approaches, using
recommended textbooks, devising a work schedule and by attending content-driven
knowledge workshops. In section 4.7.2, it was discovered that the participating teachers
utilised both procedural and conceptual knowledge approaches, construction skills, extra
tutoring, examples drawn from familiar real life situation, and additional class exercises in
the form of drill and practice in the teaching of statistical graphs. By employing these
instructional skills and strategies for teaching statistical graphs, the teachers may have
developed more knowledge of the instructional skills and strategies for teaching school
statistics. And in section 4.7.3, the individual participating teachers developed their
knowledge of learning difficulties through analysing and monitoring learners’ responses to
classwork, homework and assignments to identify such difficulties in statistical graphs. The
teachers may have also developed further knowledge of these difficulties by tackling these
using additional tutoring, extra class activities, post-teaching discussions, re-teaching, and
further explanation of the lessons they taught, individually to learners during and after the
lessons.
4.8
Summary of chapter
In this chapter, the data collected with the instruments mentioned in section 4.1 were
presented and analyse in order to determine how the participating teachers developed their
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assumed PCK in statistics teaching. The results of the qualitative data collected with the
conceptual knowledge exercise and concept mapping were analyse in order to select the
participants for the second phase of the research and determine the teachers’ content
knowledge of the statistics curriculum respectively. The lesson observations of the four
participating teachers were analysed and discussed in detail in order to tease out how they
demonstrate the PCK they have during classroom practice. The video records were used to
triangulate the data collected during the lesson observations. The teacher interview,
questionnaire, written reports and documents analyses were analysed by categorising the
responses of the participating teachers according to the theme of the study. The chapter
concluded with a highlight of how the teachers developed their assumed PCK were
determined with a summation of their subject matter content knowledge, pedagogical
knowledge and knowledge of learners’ preconceptions and learning difficulties.
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CHAPTER 5
5.0
DISCUSSION OF RESULTS
5.1
Introduction
The results of the study are discussed in this chapter. The similarities and differences in the
ways in which the participating teachers develop their pedagogical content knowledge (PCK)
in teaching statistics are examined.
The discussion begins by highlighting the research questions about teaching school statistics.
The following four components of PCK were used as the theoretical framework: (1) subject
matter content knowledge, (2) pedagogical knowledge (instructional skills and strategies), (3)
learners’ conceptions (preconceptions and misconceptions), and (4) individual learning
difficulties in the topics investigated. Pedagogical content knowledge in statistics teaching
represents a category of knowledge that teachers need to have assimilated in order to teach
the subject effectively.
These research questions were:
1
What subject matter content knowledge of statistics do mathematics teachers who
are considered to be competent have and demonstrate during classroom practice?
2
What instructional skills and strategies do these teachers use in teaching statistics?
3
What knowledge of learners’ preconceptions and learning difficulties, if any, do
they have and demonstrate during classroom practice?
4
How do these teachers develop their PCK in statistics teaching?
Components (1) and (2) above were used to answer research questions 1 and 2. In the third
component, the learners’ preconceptions and learning difficulties were identified and
discussed in order to understand how the teachers acquired their knowledge in teaching
statistics. The fourth research question was discussed as an amalgam of the key findings for
the other PCK components.
The assumed PCK profiles of the participating teachers were examined in order to determine
the similarities and differences, if any, in the ways in which the teachers develop their PCK in
school statistics teaching.
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The chapter concludes with a detailed discussion of how the results of the study provide
insight into the way in which teachers who are reputed to be competent in teaching school
mathematics develop their PCK in school statistics and evaluation of the theoretical
framework.
5.2
5.2.1
Teacher development of PCK
Teacher A
Teacher A was observed teaching histogram construction and box-and-whisker plots in a
step-wise fashion (ref Section 4.5.1: first lesson observation, and line 9; second lesson
observation, and line 5a), using the recommended mathematics textbooks and work schedule.
He started the lesson by asking the learners to name orally components of measures of central
tendency such as modes, medians and means of ungrouped data (ref Section 4.5.1, first lesson
observation, and line 1) in an attempt to determine their prior knowledge of histogram
construction. The components of measures of central tendency having been identified, the
teacher and learners prepared a frequency table from the raw data (ref Section 4.5.1, first
lesson observation and line 4a). Using this table, the histogram was constructed by first
drawing its horizontal and vertical axes. The axes were labelled with data values on the
horizontal axis, and frequencies on the vertical axis. A scale was chosen by the teacher, who
stated that the highest and lowest values of the frequencies and data values, as well as the
dimensions of the graph paper provided, had been considered (ref Table 4.5.1a). Next, the
bars of the histogram were drawn by joining the line of best fit (ref Figure 4.5.1a). Teacher
A’s lesson showed that he had adopted a rule-oriented procedural approach to teaching
histogram construction.
In teaching the construction of histograms, he gave further evidence of using more procedural
knowledge, focusing primarily on rules and algorithms, than conceptual knowledge. The
procedural approach requires simply plugging the data into the appropriate formulae, and
then working out the correct values of the quartiles for the box-and-whisker plots (ref Section
4.5.1, second lesson observation, and line 4). The most challenging aspect for this teacher
was knowing how to move from an algorithmic stage to a conceptually meaningful one as far
as the students’ learning was concerned.
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However, he used a conceptual teaching approach during the lesson and demonstrated the
mathematical connections and relationships between ogives and box-and-whisker plots by
describing how quartiles were obtained from the ogive and used in the construction of the
box-and-whisker plot (ref Section 4.5.1, second lesson observation, and line 8cii). The
relationships between the ogive and box-and-whisker plot, the calculation of the first, second,
and third quartiles, and the description of the number line on which the box-and-whisker was
drawn, with its mathematical connections, were elucidated during his lesson. A conceptualbased instructional approach endeavours to provide the reasons that make algorithms and
formulae work (Peal, 2010). The emphasis is placed on the learners’ understanding of the
relationships and connections between important statistical concepts such as the use of
quartiles to construct the box-and whisker plots on a number line (ref Figure 4.5.1c). Overall,
Teacher A implemented more of a rule-oriented procedural knowledge approach in teaching
histogram and box-and-whisker plot construction than a conceptual one. What can be
surmised from this is that he did use both knowledge approaches except, of course, that one
was dominant.
Interestingly enough, through the non-verbal cue of nodding their heads, the learners seemed
to grasp the lesson on histogram construction through the use of conceptual knowledge better
than when Teacher A adopted a rule-oriented approach. This observation was illustrated by
the fact the learners were able to answer questions involving recall and application of
procedures posed by him in order to assess how well they had understood the lesson on
histogram construction. In answering the question how do you calculate the percentage of
learners in the age group of 26–40?, learners first of all calculated the number of learners,
divided by 27 and multiplied the result by 100 to get the percentage of learners within that
age group. (ref Section 4.5.1, first lesson observation, and line 20). In the explanation, based
on his conceptual knowledge, he demonstrated his PCK in a manner that enhanced learners’
comprehension of histogram and box-and-whisker plot construction.
During the lesson, a few of the learners experienced learning difficulties such as being
uncertain about choosing a scale for labelling the data axis of the histogram (ref Section
4.5.1, first lesson observation, and line 11). The teacher identified such difficulties as being
due to lack of comprehension on the part of the learners (ref Section 4.5.1, first lesson
observation, and line 22a).
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Teacher A’s preference for the use of procedural knowledge in teaching histograms was
confirmed in the learners’ workbooks (document analysis). It was discovered that the learners
had written down the teacher’s rules or steps on how to construct histograms and box-andwhisker plots, as well as the diagrams of histogram and box-and-whisker plot (ref appendix
21, learner workbooks). Teacher A might have adopted the use of procedural knowledge
because the construction of histograms, which demands that specific procedural rules must be
followed, is consistent with a conceptual understanding of the term. In a study conducted by
Flockton, Crooks and Gilmore (2004) and Leinhardt et al (1990) on graphing, they stress that
the construction of graphs requires the sequence of drawing the axes, choosing the scale,
labelling the axes, plotting the points, and joining the lines of best fit. The order of steps, in
the case of Teacher A, demonstrated the knowledge and skills required for histogram
construction.
As observed, the learners experienced learning difficulties, particularly in labelling the data
axis with incorrect scale, which could mean that he possibly presented his lesson in a limited
way, that is, solely procedurally, without providing the reasons underlying these procedures
and clarifying the relationship between concepts (a conceptual knowledge approach) in
histogram construction (ref Section 4.5.1, first lesson observation, and line 12a). The teacher
omitted a detailed description of how to choose a scale of given data before labelling the data
axis. He merely stated the scale and used it to demonstrate the construction of a histogram.
During classwork, the learners tried to draw a histogram, which could not be accommodated
on the graph paper provided because they scaled the data axis incorrectly (ref Section 4.5.1,
Figure 4.5.1c).
It may be said that Teacher A’s PCK in terms of subject matter content knowledge
presentation did not always reveal the required variety of ways of presenting the data
handling topics to his learners for ease of access. In some instances, he demonstrated the use
of both procedural and conceptual knowledge in teaching histograms and box-and-whisker
plots, but he predominantly used a set of algorithms to demonstrate graph construction. In the
main lesson on histogram and box-and-whisker plots, he displayed factual knowledge,
procedural proficiency and conceptual understanding of the data handling topics that were
taught.
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Gersten and Benjamin (2012) note that the use of different strategies for teaching
mathematics helps to anchor the learners behaviourally and mathematically, avoids possible
learning difficulties, and achieves effective learning. This finding conforms with the
suggestion being made here, based on this study’s results, that teachers’ flexibility or the
ability to use a variety of instructional approaches (both conceptual and procedural
knowledge) should make data-handling concepts (which are said to be difficult for learners to
grasp) more meaningful and accessible to more learners. Teacher A can thus be said to have
possessed and demonstrated the required knowledge of histogram and box-and-whisker plot
construction.
Grouping method was also used as an instructional strategy for teaching the construction of
ogive by teacher A in order to provide interactive engagement , collaborative learning and to
ensure sustainability of interest in learning statistics among the learners. Learners work in
groups of four to five to calculate the quartiles of an ogive for constructing box-and-whisker
plots. The use of grouping method to sustain learners' interest in learners was given an
empirical support by Adodo and Agbayewa (2011) who report that effective classroom lesson
is achieved using grouping method for teaching. Adodo and Agbayewa (2011) further noted
that grouping method allows the teacher to better tailor the pace and content of instruction to
learners’ ability level and needs and easy management of the classroom is achieve especially
in the homogeneous grouping which teacher A adopted.
With regard to his pedagogy, Teacher A often used examples that are familiar to learners for
teaching data handling. Using the mark distribution of learners’ performances in an English
examination, he described in a step-by-step fashion how ogives are constructed, and how
quartiles are obtained and used to construct the box-and-whisker plot (ref Section 4.5.1:
second lesson observation, and line 8a). The use of familiar examples and contexts by
Teacher A is consistent with the approaches used by other workers to make the topic more
meaningful and accessible (Ball & Bass, 2000; Meletiou-Mavrotheris & Stylianou, 2002).
For example, Meletiou-Mavrotheris and Stylianou (2002) used familiar situations as
examples in the context of teaching statistics in order to improve learner access and
comprehension. According to these researchers, the teaching of rules alone (algorithmic
teaching) does not always convey meaningful relationships between the mathematics
knowledge taught in class and daily life situations (Meletiou-Mavrotheris & Stylianou, 2002).
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So this disconnects with and is seen to obscure the relevance of statistics teaching and
mathematics education in general.
Teacher A’s knowledge of learners’ preconceptions of the statistics lessons observed was
derived largely from what transpired in the classrooms, notably through his analysis of
learners’ responses to teacher classroom questions (oral probing questioning and preactivities) and classwork or assignments. During the lessons, Teacher A was able to identify
some of these difficulties or inaccurate conceptions – such as the learners’ inability to select
appropriate scales for labelling the data axis of the histogram correctly through monitoring
learner activity and questioning (ref Section 4.5.1, first lesson observation, and line 21;
Figure 4.5.1c).
In the lessons observed, for instance, the teacher did not display evidence of anticipating
learners’ potential difficulties with any of the topics. The teacher went into the lessons
without necessarily having prior knowledge or expectations of the type and nature of learning
difficulties that his learners were likely to have in teaching histogram construction. For
example, at the beginning of the lesson on histogram construction, Teacher A requested
learners to define mode, median and mean. The learners did so efficiently, based on
knowledge that they had been taught (ref Section 4.5.1, first lesson observation, and line 2).
Thus the teacher detected learners’ previous knowledge instead of preconceptions. Since the
teacher could not identify their preconceptions of histogram construction, learners were likely
to experience misconceptions and learning difficulties such as constructing a bar graph
instead of a histogram because of their poor background in scaling. Teacher A can therefore
be said to have displayed insufficient PCK in terms of the knowledge of learners’
preconceptions of histogram and box-and-whisker plot construction.
Teacher A could have addressed possible learning difficulties before or during the lesson if
he had had sufficient knowledge of learners’ preconceptions of histogram construction. When
asked in the questionnaire about his expectations of learners’ difficulties, he said merely that
there were no major problems, but he would deal with these when the learners asked him (ref
Appendix xx, item 10). The insufficiency or inadequacy of his PCK in terms of his insight
into learners’ preconceptions was a knowledge deficit that was common to all the four
teachers that were studied. The finding justifies further investigation into the reasons that
teachers, in spite of many years of teaching experience, do not seem to give much thought to
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possible misconceptions or alternative frameworks their learners are likely to bring with them
when they first come across new topics.
Penso (2002) noted that learners’ thinking about and prior knowledge of a topic is an
important aspect that should be taken seriously into consideration during teaching as it helps
to avoid possible learning difficulties that learners may encounter during the lesson. Penso
(2002) suggested that during their lesson planning, practising teachers should be encouraged
to explore varieties of instructional strategies that could elicit learners’ thinking and prior
knowledge of the concept being taught in order to be able to deal with their learning
difficulties effectively. Hill et al (2008) note that the sequence of teaching and learning may
be distorted if learners’ preconceptions are not identified in order to address learning
difficulties that learners are likely to encounter during teaching.
Teacher A addressed the learning difficulties through individual after-lesson or post-teaching
discussions, including additional exercises that were given as homework (ref Section 4.5.1,
first lesson observation, lines 23a and 23b). In his interview and written reports (ref Sections
4.7.2) the teacher confirmed the use of oral questioning, classwork and homework
assignments as strategies that he purposefully uses to evaluate how well learners have
understood the lesson and to gain insight into their pre-existing knowledge of histogram and
box-and-whisker plot constructions.
In sum, Teacher A used several instructional strategies of oral questioning, group work, using
contexts and examples familiar to learners to introduce a topic, checking and marking
learners’ classroom and homework assignments, as well as using content-specific ruleoriented graphing skills (drawing axes, choosing scale, labelling axes, plotting points and
joining line of best fit) for constructing histograms. By identifying learners’ learning
difficulties, using diagnostic questioning and monitoring techniques (already indicated),
Teacher A can be said to have used effective pedagogical strategies to elicit learners’
difficulties. But these monitoring strategies were not usually followed up with probing
questions to determine the sources of difficulty or of incorrect preconceptions.
From the discussion so far, the question is how Teacher A developed his PCK. Specifically
Teacher A’s PCK on the construction of histogram and box-and-whisker plots could be said
to have been developed over time through a series of teaching and learning experiences. It
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would be useful to identify and briefly discuss the sources of such experiences. First, in terms
of his formal education, Teacher A received further training in the teaching of mathematics
after his initial teacher training programme. He holds a BEd degree, majoring in mathematics
education, and has an Advanced Certificate in Education, specialising in teaching
mathematics and science. His qualifications may be part of the reason that his content
knowledge of the subject matter can be considered adequate. In his teaching he demonstrated
a good grasp of the various topics of histograms and box-and-whisker plots related to school
statistics.
Teacher A has 21 years’ mathematics teaching experience. Over the years his pedagogy or
instructional strategies in teaching statistics would have involved lesson planning based on
the recommended work schedule and textbooks in school statistics, delivery of lessons based
on his teaching philosophy, learned skills and feedback from his learners. Other sources of
development would have included reviews of his teaching portfolios and learners’
workbooks. All of these activities would have contributed to the development of topicspecific PCK in statistics teaching.
Teacher A attended workshops arranged by his educational district office. Most of these
workshops dealt with aspects of how to teach various mathematics topics that are considered
difficult to learn, such as data handling, analytical geometry and trigonometry. It would
appear, however, that the workshops barely considered facets of teacher knowledge of
learners’ preconceptions and sources of learning difficulties in data handling. But if they did,
the teacher did not demonstrate their potential usefulness in planning his lessons. Teacher A
appears to have limited knowledge of learners’ preconceptions that could have been used in
teaching on learners’ behalf.
In summary, Teacher A may have developed his pedagogical content knowledge from the
formal initial teacher education programme that he received; the further training obtained at
the completion of his tertiary education; attendance at in-service training workshop
programmes; periodic reviews of his own lessons and learner workbooks; and feedback over
his many years of mathematics teaching.
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5.2.2
Teacher B
Teacher B planned and taught his statistics lessons on bar graphs and ogives from the
recommended mathematics textbooks and work schedule (ref Section 4.5.2, second lesson
observation and line 9). He used a predominantly rule-driven formal procedural approach to
statistical graphs (ref Section 4.5.2: second lesson observation, lines 6a and 6b; and section
4.5.2, first lesson observation, lines 3a, 3b, 3c and 4a). As observed, in starting his lessons he
tried to identify learners’ prior knowledge of the new topic. For instance, he introduced bar
graph construction and interpretation with a pre-activity (ref Section 4.5.2: first lesson
observation, and line 1) that assessed learners’ understanding of the way in which to prepare
a frequency table. His use of pre-activities as diagnostic strategies to identify learners’ preexisting knowledge was also attested to in his responses to the teacher questionnaire and
written reports (ref Sections 4.7.3).
Teacher B taught graphical constructions of bar graphs and ogives according to the learning
outcomes of data handling as stated in the mathematics curriculum (DoBE, 2010) (ref Section
2.2). These outcomes require that learners should be able to use appropriate measures of
central tendency and spread to collect, organise, analyse, and interpret data, in order to
establish statistical and probability models for solving related problems (DoE, 2007). Teacher
B followed precisely the order in which the learning outcomes were stated in teaching his
learners how to construct bar graphs and ogives. In practice, this meant, as observed in his
lesson, drawing the axes, choosing the scale, labelling the axes, plotting the points, and
joining the line of best fit, in that order (ref Section 4.5.2, first lesson observation, lines 3a,
3c, 4a, 4c and 5). Teacher B demonstrated his PCK for drawing bar graphs in line with the
sequence described. Flockton et al (2004) confirm that for a person to understand a graph, he
or she should be able to use the construction skills of drawing the axes, labelling the axes,
plotting the points, and joining the line of best fit to construct a graph.
Teacher B’s assumed PCK on bar graphs and ogive constructions could be characterised as
procedural in terms of his lesson planning and teaching approach. Teacher B’s predominant
use of a formal procedural approach was also triangulated in the analysis of his learners’
workbooks (document analysis). The learners drew the bar graph and wrote down the
teacher’s steps on how to construct bar graphs and ogives (ref Appendix xxi; learners’
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workbooks). Teacher B might have been influenced to adopt a formal procedural approach
because of the learning outcomes of data handling as laid down in the Curriculum and
Assessment Policy Statement (CAPS) (DoBE, 2012). Besides, the construction of bar graphs
and ogives demands specific procedural rules (Flockton et al, 2004 and Leinhardt et al,
1990).
Having said that, when the teacher merely taught them the rules for constructing bar graphs,
some learners experienced certain misconceptions, confusing bar graphs with histograms, and
histograms with ogives (ref Section 4.5.2: first lesson observation, and line 9; second lesson
observation, and line 7a). A histogram is usually used to display continuous data. The horizontal
axis shows class intervals, and there are no gaps between the bars. The area of each bar shows the
frequency for the class interval. Teacher B can be said to have presented his lesson in a limited
way with insufficient explanations of how to choose the scales of grouped data (consisting of
histogram, frequency polygon, ogive, scatter plot) that are used to analyse and interpret large
data. Further, Teacher B seems not to have the flexibility to present the topics to the learners
in different ways because his lessons were presented solely according to the procedural
knowledge approach.
A detailed description of the construction of bar graphs and ogives using a conceptual
knowledge approach would have been ideal in presenting the lesson and would have avoided
possible misconceptions and learning difficulties that the learners might have encountered in
the lesson. Conceptual knowledge involves understanding mathematical ideas and procedures
and includes basic arithmetic facts (Engelbrecht, Harding & Potgieter, 2005). It is rich in
relationships among important mathematical concepts such as calculating the quartile
positions and locating the quartile itself on the ogive, class intervals and boundaries,
frequencies and cumulative frequencies of an ogive. But Teacher B’s teaching of bar graphs
and ogives was dominated by a procedural knowledge approach, which involves following a
rule or procedure without a detailed explanation of the relationships and mathematical
connections between the concepts being learned, such as calculating a quartile position and
locating it in an ogive. Thus, the teacher is probably unable to present his lesson in a variety
of ways to ensure better comprehension and understanding. A detailed description of the
concepts and their relationships, and the mathematical connections between these concepts
and even existing ideas, may help to avoid possible misconceptions and learning difficulties
that learners are likely to encounter during and after the lessons.
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Baker et al (2001) and Bornstein (2011) note that a teacher who is unable to present
mathematics content to learners in a variety of ways tends to expose them to learning
difficulties, such as constructing a histogram instead of an ogive because of the use of an
incorrect scale for labelling the data axis. A combination of procedural and conceptual
knowledge approach would have helped to deepen learners’ understanding and would have
avoided misconceptions and learning difficulties that learners might develop during the
lesson, as suggested by Engelbrecht, Harding & Potgieter (2005).
Teacher B often used familiar situations as examples for teaching data handling (ref Section
4.5.2: first lesson observation, lines 1 and 11). For instance, he described how a bar graph is
constructed using a frequency table prepared by the learners from the raw scores obtained by
learners in a mathematics test (ref Section 4.5.2, first lesson observation, and line 1). In his
lesson on bar graphs (as explained earlier) he demonstrated the construction of bar graphs
using a procedural knowledge approach. The use of familiar contexts is consistent with the
recommendations of Meletiou-Mavrotheris and Stylianou (2002), who employ everyday
situations as examples in order to make the topic accessible and meaningful to more learners.
Although Engelbrecht et al (2005) suggest that a procedural knowledge approach could help
learners to understand important demanding rule-oriented concepts, they affirmed that the use
of both procedural and conceptual knowledge would be more effective and would create
greater opportunities for improving learners’ conceptual understanding of mathematics
during the lesson (Engelbrecht et al, 2005; and Star, 2002).
During the lesson, Teacher B identified the learners’ inability to label the data axis of the
histogram correctly (ref Section 4.5.2: second lesson observation, and line 7b) by monitoring
and analysing their responses to classwork. In one example, the learners chose the scale of
grouped data and labelled the axes for data values incorrectly (ref Section 4.5.2: second
lesson observation, line 7b). Teacher B addressed such learning difficulties through extra
class activities in the form of drills and practice, as well as individual post-teaching
discussions after formal classes (ref Section 4.5.2: second lesson observation, lines 9 and 12).
The use of classwork and homework to evaluate how well learners had understood the lesson
was confirmed in the teacher’s responses to the questionnaire and written reports (ref
Sections 4.7.3).
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In terms of his knowledge of learners’ learning difficulties, Teacher B was able to detect the
misconception and learning difficulty of drawing a histogram instead of an ogive (ref Section
4.5.2: second lesson observation and line 7b). This misunderstanding could have been
because of insufficient explanation of the construction of bar graphs and ogives via the
procedural knowledge approach. As explained earlier, these learning difficulties were
discovered while monitoring and analysing the learners’ responses to classwork on bar graph
and ogive construction (ref Section 4.5.2: first lesson observation, and line 9; second lesson
observation, and line 10a). These problems were addressed by re-demonstrating the
construction of bar graphs and by using extra class activities in the case of the ogive (ref
Section 4.5.2: first lesson observation and line 10; second lesson observation, and line 9). The
teacher interview, questionnaire, written reports and teacher’s portfolios confirmed that
learners had difficulties with the construction of graphs of grouped data such as the ogive (ref
Appendix xvii, item 14; Appendix xx, item 10; items 1 and 2; and Appendix xxi, teacher
portfolios).
In terms of his knowledge of learners’ conceptions (preconceptions and misconceptions) in
statistics teaching, Teacher B tried to identify them from pre-activities and oral probing
questioning. Learners demonstrated previous knowledge of frequency tables and how data is
represented by preparing the frequency table efficiently and explaining the way in which data
is represented, but the strategy that was adopted failed to elicit learners’ preconceptions of
bar graph construction. In other words, the teacher therefore displayed insufficient knowledge
of the learners’ preconceptions of bar graphs and ogives. Learners are likely to experience
misconceptions and learning difficulties, such as an inability to label the data axis due to
incorrect scaling during the construction of ogive (ref Section 4.5.2, second lesson
observation, line 7b) when the procedural knowledge approach was adopted to teach ogive
construction. Teacher B would have been able to tackle this learning difficulty had the
learners’ preconceptions had been detected at the beginning of the lesson. When asked what
learning difficulties did the learners experience during the lesson? (ref Appendix xx, item 6),
he indicated that learners could not choose a scale of grouped data, revealing that their
learning difficulties may have emanated from the teacher’s insufficient knowledge of
learners’ preconceptions.
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Inadequacy of knowledge of learners’ preconceptions appeared to be common to all four
teachers observed during the case study period. This finding points to a further investigation
into the reasons that teachers with so many years of experience do not possess the knowledge
of learners’ possible preconceptions that may be necessary for effective teaching in the
topics. Hill et al (2008) note that the sequence of teaching and learning may not lead to easy
understanding of a concept and may not permit effective teaching if learners’ preconceptions
are not detected at the beginning of the lesson. Penso (2002) opines that teachers should
consider several opportunities to detect learners’ prior knowledge of a topic in their planning
so that the anticipated learning difficulties can easily be addressed during lesson planning and
presentation. This is an important agenda for inclusion in mathematics teachers’ education
programmes to ensure continuous improvement of PCK in statistics teaching.
How then does Teacher B develop his PCK in statistics teaching? In terms of his formal
education, Teacher B received further training in the teaching of mathematics and statistics.
He holds a BSc degree, majoring in mathematics and statistics. His qualifications may have
contributed to his content knowledge of the subject matter which can be considered adequate.
In his teaching he did not demonstrated sufficiently a good grasp of the various topics of bar
graph and ogive construction related to school statistics because his teaching was dominated
with a procedural knowledge approach that resulted to more questions from the learners
during and after the lesson seeking for clarity of the misconceptions and learning difficulties
they have encountered.
Teacher B has 10 years’ mathematics teaching experience. Within these years of teaching, his
pedagogy or instructional strategies in teaching statistics would have involved lesson
planning based on the recommended work schedule and textbooks in school statistics,
delivery of lessons based on his teaching ideology, learned skills and learners’ responses to
class activities in statistics. The review of his teaching portfolios and learners’ workbooks
were other sources for PCK development. All of these activities would have contributed to
the development of topic-specific PCK in statistics teaching.
Teacher B attended workshops organised by his educational district office. As in the case of
Teacher A, most of these workshops dealt with aspects of how to teach various mathematics
topics that are considered difficult to learn, such as data handling, analytical geometry and
trigonometry. The workshops sometimes appeared not to consider different aspects of teacher
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knowledge of learners’ preconceptions and sources of learning difficulties in statistics
teaching. But if they did, the teacher did not demonstrate their potential usefulness of the
workshop in planning his lessons as the participating teachers were unable to demonstrate
their knowledge of learners’ anticipated learning difficulties of statistical graphs. Teacher B
appears to have limited knowledge of learners’ preconceptions that could have been used in
teaching bar and ogive construction.
In summary, Teacher B may have developed his pedagogical content knowledge from the
formal initial teacher education programme that he received, attendance at in-service training
workshop programmes, periodic reviews of his own lessons and learner workbooks; and
learners’ responses to class activities in bar graphs and ogives construction.
5.2.3
Teacher C
During classroom practice, Teacher C taught his planned lessons on ogives and scatter plots
as laid out in the work schedule (DoBE, 2010). He used the recommended and supplementary
mathematics and statistics-related textbooks as sources of information for planning and
teaching his lessons on data handling (statistics) (ref Section 4.5.3, first lesson observation,
and line 16). Teacher C also displayed evidence of a procedural rather than a conceptual
knowledge approach to teaching the construction of ogives and scatter plots (ref Section
4.5.3, first lesson observation, lines 3a, 3c and 4). Teachers need to possess good
understanding of both conceptual knowledge and procedural knowledge of mathematics to be
able to provide learners with clear explanations (Engelbrecht et al, 2005 and Star, 2002).
Schneider and Stern (2010) view conceptual knowledge as mastery 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 can be viewed as consisting of rules
and procedures for solving mathematics problems. Procedural knowledge in mathematics
allows learners to solve problems quickly and efficiently because to some extent it is
automated through drill work and practice.
Teacher C demonstrated the requisite knowledge of and skills for constructing ogives in a
step-by-step manner (ref Section 4.5.3, first lesson observation, and line 4) and scatter plots
(see Section 4.5.3, second lesson observation, and line 4di). In his teaching, he moved from
the algorithmic to the conceptually meaningful stage. He began his lesson on ogives and
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scatter plots by identifying the learners’ prior knowledge of the concept of ogives through
oral questioning, and the accuracy of the homework on histograms that had previously been
taught (ref Section 4.5.3, first lesson observation, line 2bi; second lesson observation, and
line 1). Subsequently, using a cumulative frequency table prepared by the learners, an ogive
was constructed by first drawing its horizontal and vertical axes (ref Section 4.5.3, first lesson
observation, and line 4). The data values were labelled on the horizontal axis (the upper class
boundaries), and the cumulative frequencies on the vertical axis. A scale was chosen by the
teacher, who indicated that he had chosen it by considering the highest and lowest values of
the frequency and data values. The points were plotted and the line of best fit was joined to
produce the ogive (ref Section 4.5.3, first lesson observation, line 9b).
This process of constructing an ogive from grouped data depicted a rule-oriented procedural
approach. His procedural knowledge in teaching ogives (which was understandable to his
learners) is believed to have been developed as a result of his five years’ mathematics
teaching experience, using the recommended lesson plan and work schedule of the
Department of Education (DoE, 2010). The same procedural approach was used to teach
scatter plots (ref Section 4.5.3, second lesson observation, and line 4di). To demonstrate the
construction of a scatter plot, the teacher followed an algorithmic approach with
progressively less conceptual knowledge. That is, the teacher’s lesson was dominated to a
large extent by a procedural knowledge approach rather than by conceptual knowledge. Some
of the factors that may have contributed to Teacher C teaching scatter plots in a step-wise
manner, following a particular order or sequence, could be attributed to the way in which the
learning outcome of data handling is stated in the mathematics curriculum (DoBE, 2010).
The document indicates that competency in graphing requires that the learner is able to
construct, analyse, interpret statistical and probability models to solve related problem. The
construction of graphs, as stated, entails scaling, drawing axes, labelling the axes, plotting
points, and joining the line of best fit (Flockton et al, 2004; Leinhardt et al, 1990). Teacher C
followed this sequence for teaching scatter plots (ref Section 4.5.3, second lesson
observation). In the lessons observed, the teacher gave a full explanation of how to construct
a scatter plot before demonstrating how to analyse and interpret it. The learners did their
classwork in groups. They were presented with exercises on scatter plots, and were requested
to analyse and interpret the plots to determine whether there was a correlation between the
variables X and Y (ref Section 4.5.3, second lesson observation, lines 3a and 4a).
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Teacher C’s preferred procedural approach to teaching the topic was confirmed in the
learners’ workbooks, portfolios and teacher’s written reports (see appendices xx and xx1).
Owing to the limited use of the conceptual knowledge approach rather than the procedural
one – namely knowledge of the core concepts and principles and their interrelations in
teaching ogive and scatter plots – it did not come as a surprise that some learners displayed
certain misconceptions and learning difficulties in their analysis and interpretation of scatter
plots (ref Section 4.5.3, second lesson observation, and line 6). For example, a negatively
correlated linear scatter plot was interpreted by the learners as having no correlation because
of an outlier that lay far from the line of best fit (ref Section 4.5.3, second lesson observation,
lines 6 and 7). This misconception could be attributed to the rule-oriented approach that had
been adopted to describe the construction of scatter plots (ref Section 4.5.3, second lesson
observation, and line 4di), which did not allow for sufficient explanation of the
interrelationships among the data values, frequencies, lines of best fit and outliers. The
learning difficulty of interpreting a negatively correlated scatter plot as having no correlation
owing to outliers may further indicate that in teaching the construction of scatter plots the
teacher did not explain an outlier, line of best fit, type and nature of correlation, and how the
presence of an outlier affects the correlation of the X and Y variables of the scatter plot.
What can be gleaned from the discussion so far is that teachers need to possess deep
conceptual understanding of the mathematics concept that they are teaching and must be able
to illustrate why mathematical algorithms work and how these algorithms could be used to
solve problems in real-life situations (Nicholson & Darnton, 2003). The learning difficulties
experienced by the learners were subsequently addressed by Teacher C during post-activity
discussions (instructional strategy). This strategy was frequently used by Teacher C (ref
Section 4.5.3, second lesson observation, line 12) during his lessons on ogives and scatter
plots.
An important task of any teacher is to attempt to transform the content to be taught in such a
way as to make it comprehensible to the learners (Mohr & Townsend, 2002). Teacher C also
displayed evidence of a conceptual approach by providing the reasons that make the
algorithm and formula work, and by explaining the relationships between important statistical
concepts, as well as the mathematical connection between them during the lessons on ogives
(ref Section 4.5.3, first lesson observation, lines 13b and 14). It was significant that more
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learners seemed to possess a better grasp of the topic in that they were able to construct and
interpret ogives by means of this approach rather than the procedural approach (ref Section
4.5.3, first lesson observation, and line 14). In the particular lessons observed, Teacher C
explained the mathematical connections and relationships between quartile positions and the
quartiles and how quartiles can be used to interpret ogives (ref Section 4.5.3, first lesson
observation, lines 13b and 14). In doing so, Teacher C could be regarded as having displayed
progressively adequate PCK.
In his pedagogy, Teacher C used activities from everyday-life situations as examples (ref
Appendix xxi, learner workbook). For example, he demonstrated how to construct a scatter
plot using the frequency distribution table of the ages of persons infected with HIV/AIDS in
two towns (ref Section 4.5.3, second lesson observation, line 9). This use of examples drawn
from everyday life situation to illustrate scatter plot construction is in accordance with the
view held by Shulman (1987) and Krebber (2004) that transformation of the subject matter by
the teacher into a form that is more easily understood by the learners involves explanation
with examples and instructional selection of teaching methods that are adaptable to the
general characteristics of the learners. Teacher C may have decided to use examples drawn
from everyday-life situations because the topic is new in the curriculum and may be looking
for a more manageable way of presenting it to the learners in order to reinforce their
understanding.
Teacher C gained knowledge of learners’ preconceptions and learning difficulties mostly
during classroom practice. The results of this study show that he had limited knowledge of
learners’ preconceptions. As observed, learners revealed previous knowledge of ogives and
scatter plots from their responses to homework on these topics. For instance, at the beginning
of the lesson on scatter plot construction, he checked and marked learners’ homework on
scatter plots based on their previous knowledge of what they had been taught and corrected
some of their errors. While he was doing the corrections, he did not display any indication of
having knowledge of other anticipated learning difficulties. Instead, he presented the
correction procedurally, with no emphasis on the way in which previous errors that learners
had committed could be avoided during the lesson or subsequently. Learners’ learning
difficulties led to Teacher C having to provide corrections to the homework. This leads one to
the conclusion that he may not have considered identifying learners’ preconceptions in scatter
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plot construction. This information should have been used in planning the current lesson and
avoiding probable learning difficulties. The learning difficulties (constructing a histogram
instead of an ogive and misinterpreting negatively correlated scatter plots as having no
correlation due to an outlier) that were identified through monitoring and analysing the
learners’ responses to classwork (ref Section 4.5.3, second lesson observation, and line 4bii)
would have been taking into consideration during lesson planning on scatter plot
construction. Their inability to label the data axis due to incorrect scaling was identified by
oral questioning from the learners (ref Section 4.5.3, first lesson observation, and line 8a).
Penso (2002) opines that practising teachers should be encouraged to consider learners’
thinking and prior knowledge in lesson planning to avoid possible learning difficulties that
learners may experience during lesson. Hill et al (2008) also reported that the sequence of
teaching and learning may be altered if learners’ prior knowledge is not considered during
lesson planning and presentation. Teacher C addressed the learning difficulties by using a
conceptual knowledge approach, and reviewing the learners’ homework to reinforce their
understanding. He also conducted post-teaching discussions during and after ogive and
scatter plot construction lessons (ref Section 4.5.3 second lesson observation, and line 12).
The difficulties in terms of labelling the data axis of grouped data graph incorrectly were
confirmed through analysis of the learners’ workbooks (ref Appendix xxi, learners’
workbooks), as well as the teacher’s responses to the questionnaire and written reports (ref
Section 4.7.3) in which he indicated that he identified learners’ learning difficulties on graphs
of grouped data through analysis of their classwork, homework and assignments. The
learners, however, still followed the teacher after the lesson on scatter plot construction,
demanding clarification about misinterpretation of a negative linear scatter plot that he had
re-explained during the lesson. The teacher had evidently not addressed their learning
difficulties sufficiently, which means that in teaching the construction of scatter plots his
PCK was not comprehensive enough to cater for the learners’ learning difficulties
(Westwood, 2004). At this stage Teacher C did not exhibit enough PCK because his teaching
could not cater for all the learners’ learning difficulties in ogive and scatter plot construction.
He subsequently addressed the learning difficulties experienced by the learners (such as
misinterpreting a scatter plot because of outliers) in post-activity discussions, a strategy that
he used frequently in his lessons (ref Section 4.5.3, second lesson observation, line 13).
Capraro et al (2005) note that a competent mathematics teacher should be able to exhibit
progressively more PCK in his or her lessons since he or she has acquired more experience
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from formal education programmes and should plan his or her lessons in a way that is
designed to avoid any learning difficulty that learners are likely to encounter.
In summary, the PCK profile of Teacher C may be construed as an amalgam of the various
components of PCK, as defined earlier. His presumed PCK in teaching data-handling topics
lies in his ability to use oral questioning and homework to identify the learners’
preconceptions, as well as his use of construction skills and recommended mathematics and
statistics-related textbooks, and past Senior Certificate Examination question papers in
statistics to plan how to teach the construction of ogives and scatter plots. A combination of
procedural and conceptual approaches, as well as the use of everyday situations and examples
in teaching the statistics topics, constituted the instructional strategies that Teacher C
employed to teach ogives and scatter plots. By identifying learners’ learning difficulties
through monitoring and analysing learners’ responses to classwork, Teacher C can be said to
have knowledge of learners’ learning difficulties. But these difficulties were not always
followed up in terms of taking them into consideration when planning the next lesson in order
to identify learners’ preconceptions of the new topic.
The question that one would want to ask at this stage is how, then, do the teachers develop
their PCK in statistics teaching? Precisely, Teacher C’s PCK on the construction of ogive
could be said to have been developed through classroom practice and learning experiences
over time. In terms of his formal education, Teacher C received further training on the
teaching of mathematics. He holds a BSc degree, majoring in mathematics. His qualifications
may have informed the reason that his content knowledge of the subject matter can be
considered adequate.
Teacher C has five years of mathematics teaching experience. His instructional strategies in
teaching statistics would have involved lesson planning, using the recommended work
schedule and textbooks in school statistics, delivering lessons, and checking and marking
learners’ responses to homework. Other sources of PCK included reviews of his teaching
portfolios and learners’ workbooks. These activities may have contributed to the development
of topic-specific PCK in statistics teaching
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Teacher C attended workshops arranged by the District office of the Department of Basic
Education. Most of these workshops focused on the new topic of data handling and
particularly on how to teach it.
5.2.4
Teacher D
In Teacher D’s observed lessons, it was noted that he had planned and taught his lessons on
bar graphs and histograms using the Department of Basic Education’s mathematics work
schedule, and the recommended textbooks as sources of information (ref Section 4.5.4 first
lesson observation, and line 11). During his teaching of bar graph and histogram construction
(ref Section 4.5.4, first lesson observation, and line 2c), he gave more evidence of a
procedural approach to teaching bar graphs and histograms than a conceptual one. For
example, Teacher D taught the lesson on bar graphs in a step-by-step manner, beginning with
pre-activities to identify learners’ prior knowledge of bar graph construction, followed by the
preparation of a frequency table compiled by the learners using a familiar daily life example
(ref Section 4.5.4, first lesson observation, lines 1 and 2c). In this case, a frequency table was
prepared of the number of cars in a car park according to their make (ref Section 4.5.4, first
lesson observation, line 1). Next, with the help of the frequency table, a bar graph was
constructed by first drawing its horizontal and vertical axes and labelling them appropriately.
A scale was chosen by the teacher with the explanation that this was done by considering the
highest and lowest values of the frequencies and the companies that manufactured the cars.
Next, the points were plotted and the line of best fit was joined to produce the bar graph (ref
Section 4.5.4, first lesson observation, lines 2c and 3). The teacher’s specific strategy for
teaching bar graph construction followed a rule-oriented procedural approach using
procedural knowledge.
Engelbrecht et al (2005) describe the procedural knowledge approach as “following a rule or
procedures flexibly, accurately, efficiently and appropriately in completing a given task”. For
example, in constructing a statistical graph, procedural knowledge approach requires a series
of actions such as drawing the axes, choosing the scale, labelling the axes, plotting the points
and joining the line of best fit. But what may be sometimes challenging is knowing how to
move from the procedural stage to a conceptual meaningful one in terms of the students’
learning.
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As with the other teachers, Teacher D’s procedural knowledge may have been developed
over his 15 years of teaching mathematics in high school, using the recommended lesson plan
and work schedule for statistics (DoBE, 2010). It could be suggested that although Teacher D
possesses adequate ways of presenting bar graph construction to his learners, his PCK may be
limited in the sense that he presented his lesson procedurally, an approach that was not
always responsive to the learners’ needs. Consequently, some of the learners constructed the
classwork task without leaving spaces between the bars of the graph. The inability to consider
the consistency of spaces between the bars of a graph during lesson presentation resulted in
the learning difficulties that the learners experienced during classroom practice.
According to Shulman (1987), representation involves the teacher thinking through the key
ideas and identifying alternative ways of presenting them to the learners. It is a stage in which
suitable examples, demonstrations and explanations are used to build a bridge between the
teacher’s comprehension of the subject matter and what is required for the learners
(Ibeawuchi, 2010). Multiple forms of representations are highly desirable if one is to be
successful in the teaching process (Rollnick et al, 2008). Teacher D, however, in certain
graphing topics, did display evidence of an alternative conceptual knowledge approach in
teaching histograms (ref Section 4.5.4, second lesson observation, lines 11). Engelbrecht et al
(2005) describe the conceptual knowledge approach as “involving an understanding of
mathematical ideas and procedures consisting of the knowledge of basic arithmetic facts”.
Therefore it is knowledge that is rich in relationships and understanding of important
statistical concepts in bar graph and histogram constructions. In the lesson observed, Teacher
D explained in detail the meaning of a histogram. According to Teacher D, “a histogram is a
graphical representation, showing a visual impression of the distribution of grouped data. It
consists of tabular frequencies shown as adjacent rectangular bars, erected over discrete
intervals, with an area equal to the frequency of the observations in the interval. Unlike the
bar graph, a histogram is used to represent a large set of data (e.g. a population census)
visually, but with no spaces between the bars” (ref Section 4.5.4, second lesson observation,
lines 5b). His conceptual approach (presumably PCK) to teaching the construction of a
histogram enhanced conceptual understanding of the topic as the learners seemed to be
satisfied with Teacher D’s conceptual explanation (ref Section 4.5.4, second lesson
observation, and line 11) of how to construct a histogram after the learners had experienced
misconceptions and learning difficulties in labelling the data axis (ref Section 4.5.4,second
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lesson observation, and line 10). They displayed the non-verbal cue of nodding their heads in
agreement with the teacher’s explanation (ref Section 4,5,4, second lesson observation, and
line 12).
From the lessons observed with Teacher D, he used a procedural knowledge approach more
rather than a conceptual knowledge approach. His preferred use of this approach was
confirmed in the document analysis conducted in Teacher D’s learner workbooks. The
learners had completed the diagrams on bar graphs and histograms efficiently, with
indications of the procedures that had been adopted in constructing these statistical graphs.
Star (2002) argues that it is important for practising teachers to possess both kinds of
knowledge in order to impart teaching to the learners in a meaningful way. The use of a ruleoriented procedural approach and a conceptual knowledge approach reveals that teachers are
looking for ways of making the teaching of bar graphs and histogram comprehensible and
accessible to their learners. Moreover, the construction of graphs demands that a particular
order of actions should be followed, consistent with conceptual understanding. Teacher D can
therefore be said to possess and demonstrate the required knowledge of bar graph and
histogram construction.
Over and above this, Teacher D was able to identify learning difficulties experienced by the
learners during the lesson and alternative conceptions from the various graphing exercises
that were carried out by the learners. One such learning difficulty was their inability to
choose the correct scale for labelling the data axis. This meant that they constructed a bar
graph instead of a histogram (ref Section4.5.4, second lesson observation, and line 6a). In this
case, Teacher D may be said to have presented his lesson in a limited way. His lessons were
dominated by procedural knowledge teaching without providing the reasons underlying such
procedures. He may not have accommodated the possibility of anticipating learning
difficulties during the lessons on bar graph and histogram construction in his lesson planning
and presentation and resolving them. For instance, he indicated the scale for constructing a
bar graph and how it was obtained without explaining his reasons for choosing it, which
shows that he may have presented his lesson in a limited way. When the learners adopted the
same procedure to construct the bar graph during classwork, they did not consider the
consistency of spacing in a bar graph, which resulted in a histogram instead of a bar graph. In
terms of subject matter content knowledge, Teacher D may not have demonstrated the
required variety of ways of presenting bar graphs and histogram construction for easy
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comprehension by the learners. Gersten and Benjamin (2012) note that the use of several
strategies for teaching mathematics helps to deepen learners’ understanding behaviourally
and mathematically, avoids possible alternative conceptions and learning difficulties, and
achieves effective learning. This means that teachers should be flexible (able to use a variety
of instructional strategies to make content more accessible to more learners) in the
representation of bar graph and histogram construction.
Regarding knowledge of instructional skills and strategies, Teacher D used analysis of
learners’ responses to classwork on bar graphs and histograms to identify their alternative
conceptions and learning difficulties (ref Sections 4.5.4, second lesson observation, and line
6a). The teacher questionnaire, written report and learner workbooks confirmed this use of
monitoring and analysing learners’ responses to classwork to identify their alternative
conceptions and learning difficulties. He addressed these difficulties through the instructional
strategies of additional explanations, extra class activities, and examples related to familiar
situations. These methods are consistent with the findings of Penso (2002), Westwood
(2004), Bucat (2004), Mitchel and Mueller (2006) and Cazorla (2006), who adopted the same
strategies for dealing with learners’ misconceptions and learning difficulties. In practice,
teachers are expected to design good teaching and learning instructions that take into
consideration ways of identifying and addressing learners’ learning difficulties (Westwood,
2004; Jong et al, 2005; and Rollnick et al, 2008). The other instructional skills that Teacher D
used in teaching bar graphs and histograms were the construction skills involving drawing the
axes, choosing the scale, labelling the axes, plotting points and joining line of best fit, which
require a procedural knowledge approach.
The greater part of Teacher D’s knowledge of learners’ preconceptions and learning
difficulties was gathered while teaching the assigned topic in statistical graphs. As observed
earlier, during classwork the learners’ inability to choose an appropriate scale for histogram
and bar graph construction was identified through monitoring and analysing their responses
(ref Section 4.5.4, second lesson observation, and line 6a). But the teacher did not display any
evidence of having anticipated the learners’ learning difficulties with bar graph and histogram
construction, revealing that he may have gone into class without necessarily having
knowledge of learners’ possible learning difficulties in these constructions. To this end,
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Teacher D can therefore be said to have displayed insufficient PCK in terms of awareness of
learners’ preconceptions of bar graph and histogram constructions. From instance, at the start
of the lesson, Teacher D requested the learners to prepare a frequency table of the makes of
cars in a car park. The learners prepared the frequency table efficiently using previous
knowledge. Thus, the teacher realised that the learners had previous knowledge that could be
linked to bar graph construction, and no preconception was identified. When asked about the
learning difficulties that learners might have had or were likely to experience during his
lesson, he indicated that although that the learners had problems in determining the mid
points and constructing graphs of group data, he would deal with difficulties that might arise
(ref Appendix xx, item 10).
Inadequacy in teachers’ knowledge of learners’ preconceptions was common to the entire
group of teachers involved in this study. This suggests the need for further investigation into
the reasons that such teachers with many years of experience should have such a knowledge
deficit in an area that is essential for effective classroom practice. As indicated in the section
on Teacher A, Penso (2002) suggested that in their lesson planning, practising teachers
should explore a variety of instructional strategies that would elicit learners’ thinking and
prior knowledge of the concept being taught in order to deal with their learning difficulties
effectively. Hill et al (2008) note that the sequence of teaching and learning may by
interfered with and possibly create opportunities for learning difficulties to occur if learners’
preconceptions are not considered when planning and presenting a lesson.
Teacher D tried to address difficulties through extra explanations and homework assignments
(ref Section 4.5.4, second lesson observation, and line 13). The teacher questionnaire, written
reports and document analysis confirm the use of pre-activities, extra explanations and class
activities in the form of classwork and homework to evaluate how well learners have
understood the lessons and to gain insight into learners’ pre-existing knowledge of bar graph
and histogram construction.
It can be gleaned from the above discussion that Teacher D displayed a combination of the
components of PCK that were identified earlier (Hill et al, 2008). Teacher D’s presumed
PCK is evidenced in his lesson planning and preparation, and in the use of textbooks in
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school statistics and mathematics, as well as other learning materials, such as past Senior
Certificate Examination question papers. Pre-activities and correction of homework
assignments were the instructional strategies he used to identify preconceptions about bar
graphs and histograms. A combination of procedural and conceptual approaches to the
teaching of statistics, as well as the use of exemplars drawn from familiar situations, was
another instructional strategy that Teacher D used to teach the construction of bar graphs and
histograms (ref Section 4.5.4, second lesson observation, lines 5c). Teacher D at this stage of
using several instructional skills and strategies can be considered to have displayed
knowledge of instructional skills and strategies for teaching bar graphs and histograms.
Misconceptions such as drawing a histogram instead of a bar graph (ref Section 4.5.4, first
lesson observation, and line 8) were addressed through post-teaching discussions, additional
explanations during the lessons and homework (ref Section 4.5.4, first lesson observation,
lines 9 and 11).
In sum, the sources for the development of Teacher D’s PCK can partly be linked to the
formal education that he acquired from a teacher training programme. He holds a BEd and
SED, majoring in mathematics education. These qualifications and his 15 years of experience
may have provided Teacher D with the opportunities to develop his content and pedagogical
knowledge in statistics teaching. His instructional strategies over the years would have
involved the use of lesson planning in line with the recommended work schedule, textbooks
in school statistics and presentation of his lessons based on learning skills and reviewing of
learners’ classwork, homework and assignments. Other sources of development of his topicspecific PCK would have included reviews of his portfolios and learner workbooks. Teacher
D attended content knowledge workshops organised by the Department of Basic Education
(DoBE). Most of these workshops dealt with new aspects of the mathematics curriculum,
especially the issues around teaching topics such as data handling.
5.3
Evaluation of theoretical framework
To evaluate the theoretical framework of this study is to determine to what extent the
theoretical framework has enabled the researcher to answer the research questions.
The conceptual knowledge exercise, concept mapping exercise, teacher interviews, lesson
observations and document analysis were the instruments used to examine the subject matter
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content knowledge of the participating teachers in school statistics in this study. The intention
of the researcher in using these instruments for data collection was to determine the subject
matter content knowledge that the participating teachers demonstrated in classroom practice.
What can be gleaned from the results is that the instruments allowed the researcher to capture
the teachers’ PCK in terms of the subject matter content knowledge in statistics teaching. The
concept map exercise was used as a proxy, but was not sufficient to determine how
knowledgeable the teachers were about the contents of the curriculum (ref Section 4.4). The
teachers should have been requested to write an examination in order to determine their
content knowledge of the topic. But because it might be difficult to get the teachers to write
an examination, a concept mapping exercise was considered a good proxy for assessing their
content knowledge. Another way in which the teachers’ content knowledge could have been
examined was through certification. That is by reviewing the certificate obtained from
colleges and universities. Considering a certificate in mathematics education without
observing how a teacher demonstrates his or her content knowledge in the classroom may not
be sufficient to determine whether that teacher possesses content knowledge of a topic.
Hence, lesson observations were used to assess the teachers’ subject matter content
knowledge and how well they demonstrated this knowledge in statistics teaching. Although
Mahvunga and Rollnick (2011) suggest that a quantitative research study may be sufficient to
assess teachers’ content knowledge, their study failed to indicate how to assess the quality of
teachers’ content knowledge, which can be determined only during classroom practice. This
assertion is given wide empirical support by researchers such as Toerien (2011), Ball et al
(2008), Capraro et al (2005), Jong et al (2005), Lee and Luft (2008), Jong (2003) and GessNewsome and Lederman (2001), who all note that PCK is rooted in classroom practice. Any
research into teachers’ PCK that does not consider the use of lesson observation may fail to
fully convey the required information about how teachers develop topic-specific PCK.
Through lesson observation, it was possible to determine how the teachers demonstrated their
content knowledge of certain topics. Lesson observation provided opportunities to experience
the details, nuances and dimensions that the teachers used in their classroom practice in order
to determine the adequacy of their subject matter content knowledge (ref Sections 4.5.1–
4.5.4). Through the teacher interviews, it appears that the teachers’ educational backgrounds
that may have enabled them to develop topic-specific content knowledge in statistics were
determined (ref Section 4.7.1).
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While researchers such as Shulman (1986), Van Driel et al (1998), and Magnusson (1999)
use subject matter knowledge consisting of syntactic and substantive knowledge acquired in
formal education, in this study the subject matter content knowledge focused on the content
to be taught and learned by the students. The use of subject matter content knowledge as the
theoretical framework for this study proved useful in determining the procedural and
conceptual knowledge (component of the PCK) that a teacher demonstrates in teaching
statistical graphs. Other PCK studies (Plotz, 2007; Lee & Luft, 2008; Adela, 2009;
Ibeawuchi, 2010; Ogbonnaya, 2011; and Toerien, 2011) share the same view of using subject
matter content knowledge as a theoretical framework for examining teachers’ PCK
development in mathematics. These authors also assess the subject matter by making the
teachers write a test on the content of the topic under investigation. The instruments
developed with the framework were therefore considered adequate to determine teachers’
subject matter content knowledge in statistics teaching and the theoretical framework can be
considered adequate and valid.
The teacher questionnaire, which focused on what the teachers did while teaching the
assigned topic, and the written reports used to triangulating the data collected with lesson
observations were used to determine the pedagogical knowledge (instructional skills and
strategies) that the teachers used in teaching school statistics. Other instruments used to
assess the teachers’ pedagogical knowledge were lesson observation and document analyses.
The questionnaire revealed many aspects of the teachers’ PCK, such as knowledge of
instructional skills and strategies for teaching statistical graphs. These strategies included oral
probing questioning, checking and marking learners’ homework and pre-activities to
determine learners’ pre-existing knowledge (ref Section 4.8). The lesson observations,
teacher written reports and document analyses confirmed the use of these instructional
strategies. These activities were crucial in determining learners’ conceptions about statistical
graphs, as suggested by Krebber (2004), Westwood (2004) and Ball et al (2008), but did not
elicit learners’ preconceptions in statistical graphs. From the lesson observations, it was not
possible to determine learners’ preconceptions because the strategies the teachers adopted to
do so did not elicit them. Instead, the learners displayed previous knowledge linked to
learning the new topic. In fact, the teachers did not have knowledge of the instructional skills
and strategies that might have been necessary to determine the learners’ preconceptions in
statistical graphs.
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As Krebber (2004) and Westwood (2004) suggest, the use of the instructional strategies of
oral probing questioning, pre-activities, checking and marking learners’ responses to
classwork, homework and examining learners’ understanding, as well as identifying their
misconceptions and learning difficulties in statistical graphs, is critical in learning and could
motivate the development of teachers’ pedagogical knowledge. Loughran et al (2004), Ball et
al (2008), and Vistro-Yu (2003) regard teachers’ pedagogical knowledge as crucial to PCK
development. Having ascertained the instructional skills and strategies demonstrated by the
teachers through the teacher questionnaire, written reports, document analyses and lesson
observation, the researcher believes that the teachers’ pedagogical knowledge can be
considered a valid theoretical framework for determining the PCK required for teaching
school statistics.
However, the framework provided an opportunity to reveal that the teachers had some
knowledge of learners’ misconceptions, as individually they were able to identify
misconceptions through analysis of learners’ responses to classwork, homework and
assignments in statistical graphs. The activities of identifying and addressing learners’
misconceptions are critical aspects of teaching and learning. Penso (2002), Carzola (2006)
and Westwood (2004) note that a teacher who lacks the ability to identify and address
learners’ misconceptions may experience poor content delivery in classroom practice.
Practising mathematics teachers are encouraged to learn about the possible instructional skills
and strategies for identifying and addressing learners’ alternative conceptions in statistical
graphs.
Penso (2002), Westwood (2004) and Carzolia (2006) also posit that if learners’ alternative
conceptions and difficulties are not identified and addressed in the preparation and
presentation of lessons, negative lesson presentations can occur. The lesson observations (ref
Section 4.5.1–4.5.4), teachers’ written reports, and learners’ and teachers’ portfolios
confirmed that the participating teachers know about learners’ learning difficulties in
statistics (ref Sections 4.7–4.10). Therefore, knowledge of learners’ learning difficulties can
be considered adequate as a theoretical framework for capturing teachers’ PCK in statistics
teaching.
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5.4
Summary of the chapter
This chapter opened with a brief recapitulation of the research questions and PCK
components as a theoretical framework for this study.
The teachers demonstrated that they possess content knowledge of school statistics. However,
the predominant approach to putting across statistical ideas to their learners about data
handling, particularly the construction of statistical graphs, was procedural. A conceptual
approach was used less to some extent and not as the same degree as procedural approach.
The individual teachers are presumed to have developed their PCK in statistics teaching by
extending their knowledge of the subject matter content through formal education
programmes and the use of topic-specific mathematics and statistics textbooks and other
publications as sources for lesson planning and teaching.
The instructional skills and strategies used by the participating teachers for teaching specific
statistics topics consisted largely of oral questioning, pre-activities, and post-teaching
discussions to determine preconceptions. By using these instructional skills and strategies to
teaching statistical graphs, the participating teachers may have developed their PCK in
statistics teaching. An analysis of the learners’ classwork, homework, assignments, and postteaching discussions was used to determine where the learners’ misconceptions and learning
difficulties lay. All four teachers, although at different times, used extra tutoring, problemsolving activities involving familiar daily-life contexts, individualised teaching, post-teaching
discussions, and repetition of the lessons to address learners’ difficulties and misconceptions
(ref Section 5.2.1-5.2.4).
This chapter concludes with an evaluation of how the theoretical framework was used to
ascertain whether it provided adequate opportunities to develop instruments for collecting
data to answer the research questions.
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CHAPTER 6
6.0
SUMMARY AND RECOMMENDATIONS OF THE STUDY
6.1
Introduction
This chapter presents a summary of the study, recommendations, and suggestions for further
research. The main aim of this study was to investigate how competent mathematics teachers
whose learners perform consistently well in the Grade 12 mathematics National Senior
Certificate Examination develops pedagogical content knowledge in school statistics
teaching. Specifically, it explored what these teachers do in classroom practice when teaching
data handling topics to the learners. In addition, the study probed the implications that PCK
has for mathematics teacher education programmes (ref Section 6.5).
6.2
Focus of the study
The study investigated how competent mathematics teachers develop PCK in statistics
teaching (which has recently been introduced as a formal aspect of mathematics in the
National Curriculum Statements (now changed to Curriculum and Assessment Policy
Statements (CAPS)) in South Africa. The chief examiner’s report (DoE, 2009; DoBE, 2012)
states that learners’ poor performance in statistics may mean that mathematics teachers have
not acquired sufficient PCK for teaching the subject. In addition, delegates at the
International Commission for Mathematics Instruction and International Association for
Statistics Educators joint conferences (ICMI/IASE, 2007, 2011) attributed learners’ poor
achievement in statistics to underdevelopment of PCK by practising mathematics teachers.
This research is therefore intended to explore the manner in which competent mathematics
teachers develop PCK in statistics teaching.
A multi-method approach involving the use of several research instruments such as a
conceptual knowledge exercise, concept mapping, lesson observation, teacher questionnaires,
interviews, written reports, video records and document analysis for data collection, was
adopted to carry out the investigation. Mathematics teachers were identified who were
perceived to be competent in teaching mathematics, based on their school performance in the
senior certificate examination, together with recommendations from principals, peers and
subject facilitators.
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The research questions that guided the study are:
1)
What subject matter content knowledge of statistics do the mathematics teachers have
and demonstrate during classroom practice?
2)
What instructional skill and strategies do these teachers use in teaching statistics?
3)
What knowledge of learners’ preconceptions and learning difficulties, if any, do they
have and demonstrate during classroom practice?
4)
How do these teachers develop PCK in statistic teaching?
A qualitative research approach involving the case study method was used to collect data.
The data were analysed to determine the teachers’ assumed PCK and how they might have
developed their PCK profile in statistics teaching. PCK, in the context of this study, was used
as a theoretical framework to try to determine how they developed their assumed PCK in
statistics teaching (ref Section 1.7). It was defined as ‘an amalgam of practising teachers’
content knowledge in school statistics; their pedagogical knowledge (instructional skills and
strategies) and learners’ conceptions and learning difficulties in statistics teaching’ (Shulman,
1987 and Ball et al, 2008: 391)
In applying the PCK as a theoretical framework, certain assumptions were made, as indicated
in chapter 1, to enable the investigator to proceed with the study. These assumptions are as
follows:
•
PCK represents a category of knowledge that describes the quality of an expert
teacher (Miller, 2006).
•
PCK provides a framework that can be used to describe the origin of its critical
teacher knowledge, but not all the teachers have the same PCK (Miller, 2006).
•
PCK is a constructivist process and, therefore, a continually changing body of
knowledge (Miller, 2006).
•
PCK can be measured by conceptualising the construct and using multiple assessment
techniques, including classroom practice (Hill, 2008).
It is currently a widely accepted belief that PCK represents a category of knowledge needed
for a novice teacher to mature into an expert (Miller, 2006). Ball et al (2008) described
teacher knowledge as an amalgamation of subject matter and pedagogy. The blend of
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different forms of teacher knowledge has forced many teacher education programmes to
create new pedagogical activities that engage pre-service teachers in terms of the teachers’
classroom practice. ‘The same vision of how to improve classroom practice has provided a
focus on education research; unfortunately PCK remains a category of knowledge that is
difficult to isolate and research’ (Miller, 2006). However, the teachers’ classroom practice in
statistics teaching in the context of this study was investigated in a case study using lesson
observation to see how they demonstrated their subject matter content knowledge,
pedagogical knowledge, and knowledge of learners’ conceptions and learning difficulties.
Data gathered from lesson observation were triangulated with data collected from concept
mapping, teacher questionnaires, interviews, written reports, video records and document
analysis in order to determine how the mathematics teachers develop their PCK in statistics
teaching for learner performance and classroom practice improvement.
6.3
Summary of the results according to the theoretical framework
A summary of the results from the investigation is as follows:
6.3.1
Knowledge of the subject matter content
The four participating teachers taught statistical graphs predominantly using procedural
knowledge and less frequently as conceptual knowledge. The use of procedural knowledge
was to some extent dictated by the nature of the topic, which required learners to be able to
collect, organise, analyse and interpret statistical and probability models to solve related
problem (DoBE, 2010). A second factor that leads to the use of procedural knowledge is the
way in which statistical graphs should be constructed, which involves drawing axes, choosing
scales, labelling axes, plotting points and joining the lines of best fit. Other processes in
developing subject matter content knowledge included the frequent use of mathematics
textbooks, CAPS documents, as well as attendance at workshops (ref Appendix xvii).
6.3.2 Pedagogical knowledge (instructional skills and strategies)
Instructional skills are the most specific category of teaching behaviour. They are necessary
for procedural purposes and for structuring appropriate learning experiences for learners. In
this study, instructional skills and strategies, involving construction skills such as drawing
axes, choosing scale, labelling axes, plotting the points and joining the lines of best fit were
used in constructing statistical graphs. Instructional strategies such as oral probing
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questioning, pre-activities, pre- and post-teaching discussion were used by the individual
teachers to determine learners’ prior knowledge in statistical graph construction. Checking
and marking learners’ responses to homework were other assessment strategies that helped to
determine learners’ prior knowledge in statistical graphs and learning difficulties. Procedural
and conceptual approaches were used to describe how to construct statistical graphs such as
the bar graphs, box-and-whisker plots, ogives, histograms and scatter plots. Individual and
grouped classwork, homework and assignments as well as oral probing questioning were also
used to assess how well learners’ have understood the lessons on these graphical
constructions. An analysis of learners’ responses to classwork, homework and assignments
was the main assessment strategy that the participating teachers used to identify learners’
misconceptions and learning difficulties in statistics teaching. While some learners showed
that they had grasped what the teachers taught, a few experienced learning difficulties.
Instructional strategies such as the use of extra tutoring, class activities in the form of drill
and practice, explanation, examples drawn from familiar situations and post-teaching
discussions were used to address learners’ misconceptions and learning difficulties in
statistical graph construction.
The participating teachers claimed that the instructional skills and strategies used in teaching
statistics were developed through formal education and classroom practice (ref Section 4.7.2
and Appendix xvii). The development of instructional skills and strategies varies from
teacher to teacher, depending on the topic, feedback from the learners, and the learners’ prior
knowledge of that topic. The results drawn from lesson observation (ref Sections 4.5.1–
4.5.4), interviews and the questionnaires (ref Appendices xvii and xxviii) showed that the
participating teachers used topic-specific instructional strategy of providing exercises in
statistics in which learners were required to solve problems, while the teachers monitored and
guided them (as in classwork), which allowed learners to construct knowledge by themselves,
thereby influencing their active participation in the lessons. By using instructional skills such
as topic-specific construction skills, and the instructional strategies of oral probing
questioning, pre-activities, extra tutoring and class activities and post-teaching discussion, as
well as assessment strategies of analysing learners’ responses to written works to determine
learners’ misconceptions and learning difficulties, the participating teachers’ may have
intensified and broaden their knowledge of the instructional skills and strategies used in
teaching school statistics.
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6.3.3
Knowledge of learners’ preconceptions and learning difficulties
The most notable learning difficulty observed in the lessons of all four teachers was the
inability to construct and interpret graphs of grouped data (ref Sections 5.2.1-5.2.4). The main
challenge, in part, was owing to learners’ inability to choose an appropriate scale (ref
Sections 5.2.1- 5.2.4). Second, the learners had difficulty in labelling the axes without proper
scaling for constructing the statistical graph on the paper provided (ref Figure 4.5.1c).
The teachers developed knowledge of learners’ learning difficulties through analysis of their
classwork, homework and assignments, as well as through post-teaching discussions on
statistical graphs construction (ref Sections 5.2.1-5.2.4). Constant examination of the
learners’ workbooks helped to reinforce the teachers’ insight into learners’ conceptions
(preconceptions and misconceptions) of statistics topics (ref Sections 4.5.1–4.5.4).
The teachers addressed these difficulties at various times through extra classes, problemsolving tasks using familiar real-life examples, post-teaching discussions, and teaching on a
one-to-one basis after normal school hours (ref Section 5.3.4). The process of identifying and
addressing learners’ learning difficulties should have provided the teachers with ample
knowledge of learners’ preconception and learning difficulties in statistics teaching. But it is
surprising that after so many years of teaching mathematics, some of the teachers are not
aware of these problems. This lack of familiarity with learners’ anticipated learning
difficulties could be because the topic was recently introduced into the curriculum. Therefore,
the teachers may not have developed the required PCK for addressing the difficulties which
learners’ may experience in learning school statistics. However, by identifying and
addressing learners’ alternative conceptions and learning difficulties, the participating
teachers may have gained more knowledge of the learners’ learning difficulties in statistics
teaching.
6.4`
Concluding remarks
Based on the findings of this study, individual teachers constructed their PCK in statistics
teaching by:
•
Formally developing their knowledge of the subject matter in an accredited
formal education programme in which they had the opportunity to study the
subject matter and methodology of school statistics
•
Teaching school statistics using procedural and conceptual knowledge to some
extent (ref Sections 4.5.1-4.5.4.).
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•
Using several mathematics and statistics textbooks, past senior certificate
examination question papers in statistics and other materials in lesson
preparation, consistent with their understanding of the nature of statistics in
school mathematics and how it should be taught (ref Section 5.2.1-5.2.4). For
example, Teacher A taught his lesson of histogram construction and assigned
classwork and homework using learners’ mathematics textbook (ref Section
4.5.1, first lesson observation, and line 23b).
•
Using varied topic-specific instructional skills such as construction skills
(involving the drawing of axes, choosing of scale, labelling of axes, plotting thee
points and joining the line of best fit), problem-solving, assessment (in the form
of oral probing questioning, classwork, homework and assignments), and
interpretation skill (comprising of determining the relationship between X and Y,
and based on the relationship between X and Y values, one can say whether there
is positive correlation, negative correlation, or no correlation as in, second lesson
observation, and line 4bii), in teaching scatter plots (ref Section 4.5.3)
•
Using diagnostic techniques (oral questioning, pre-activity and class discussions)
and a review of previous lessons to introduce lessons, and to determine learners’
preconceptions in statistics teaching (ref Section 4.7.3)
•
Using a variety of assessment techniques such as classwork, homework and
assignments and grouped work in statistical graphs to assess how well learners
understood the lesson on statistical graphs and to identify their difficulties (ref
Sections 4.7.3).
•
Continually updating their knowledge of school statistics by attending content
knowledge workshops and other teacher development programmes designed to
improve content awareness and practice (ref Section 5.3)
By knowing how teachers develop PCK for teaching school statistics, teacher educators will
be able to develop greater understanding and insight into designing programmes to teach
topics that were previously included only at tertiary level.
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6.5
Educational implications of the study
Based on the results of this study, the educational implications can be summarised as follows:
The findings of this study can be used to provide a knowledge base and process to be
employed by mathematics teachers to develop PCK for the continuous improvement of
effective mathematics classroom practice. For instance, the teachers developed knowledge of
learners’ learning difficulties by analysing their responses to classwork, homework and
assignments and during pre- and post-activity discussions. Regular examinations of learners’
workbooks helped to reinforce the teachers’ familiarity with learners’ conceptions and
learning difficulties of statistics topics. Learning difficulties were generally addressed by the
teacher engaging the learners on a one-to-one basis or collectively during or after school
hours.
The development of subject matter content knowledge of statistics renders it an essential
component of PCK for teaching it at school level. When teaching statistics, teachers’ actions
were determined to a large extent by the depth of their PCK, thereby making subject matter
content knowledge an essential component of their ongoing learning of school statistics for
the improvement of their expertise in statistics and effective classroom practice.
‘Pedagogical content knowledge research links knowledge of teaching with knowledge of
learning’ (Adela, 2009). This is a powerful base on which to build teaching expertise. In this
study, formal education in mathematics was found to be a prerequisite in developing
teachers’ subject matter content and pedagogical knowledge. Several research reports have
attempted to establish how PCK is developed in science and mathematics. As PCK is topicspecific, however, little attempt has been made to determine how PCK is developed in the
context of teaching statistics by mathematics teachers. The research that is available suggests
that this type of information is meagre. This study has therefore furnished insight into how
PCK is developed by competent mathematics teachers. A detailed description was given of
examples of the PCK of mathematics teachers in terms of improving learners’ performance in
statistics and for consideration by teacher trainers in designing statistics teacher education
programmes for in-service and pre-service teachers.
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In this study as indicate in section 1.6, PCK was conceptualised to include content specific
knowledge, content specific instructional strategies and learners’ preconceptions of specific
concept, rules and skills. ‘PCK development is a complex process and it is not clear how it is
developed in statistics teaching for mathematics classroom practices (Jong, 2003). PCK is
distinct from a general knowledge of pedagogy, educational purpose and learners’
characteristics. Moreover, because PCK is concerned with the teaching of a particular topic
e.g statistics, it may turn out to differ considerably from the subject matter itself’ (Jong, Van
Driel and Verloop, 2005:948). PCK is said to develop by an iterative process that is rooted in
classroom practice. The implication is that many beginning teachers have little or no PCK at
their disposal, particularly in statistics teaching (ref Section 1.6).
From the description of the knowledge-base and process employed by competent
mathematics teachers in developing PCK in statistics teaching, notions of and insight into
PCK could be obtained that can be incorporated into a mathematics education programme for
in-service and pre-services mathematics teachers, thereby contributing to the continuous
improvement of the mathematics teacher education programme and teachers’ PCK.
6.6
Suggestions for further study
The results of this study present several areas for further research opportunities. These areas
are suggested:
•
Large-scale research needs to be conducted on the kind of subject matter content
knowledge that a teacher needs for development of PCK in statistics, especially in the
construction and interpretation of graphs of grouped data, which many teachers seem
to find difficult to teach.
•
More studies need to be conducted to determine the impact of teachers’ knowledge of
learners’ preconceptions as a theoretical framework for investigating teachers PCK in
statistics teaching.
•
This study found that procedural and conceptual knowledge were both necessary for
teaching statistical graphs, especially in addressing learners’ misconceptions and
learning difficulties. Further studies are needed to determine how well both
approaches can be applied to other aspects of school statistics.
•
More researches need to be conducted on why teachers with over five years
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experience of teaching mathematics lack sufficient knowledge of learners’
preconceptions in statistics teaching.
6.7
Limitations of the study
This study may have been influenced by these limitations, which should be taken into
consideration when interpreting the results:
•
Selection of the participants created a problem that led to having only a few in the
study. The number of participants was reduced because of the criteria used in
selection. The schools from which the participants were selected had to obtained a
pass rate of 70% and above in mathematics in the senior certificate examination for at
least two years. This left the researcher with a small number of schools from which to
select willing participants.
•
Assessment of teaching competencies is usually associated with inherent limitations
as they are coloured by personal observer idiosyncratic tendencies. The results of this
study during the lesson observations may not necessarily be replicated. The process of
interpreting teachers’ practice and decisions, and placing them into specific
pedagogical categories may not always be 100% correct. The possible errors in the
interpretations were reduced by the triangulation of data, using open assessments
(questionnaires to confirm the teacher observations and the categories assigned) and
negotiations for placing pedagogical actions into appropriate categories of how the
teachers developed their PCK in the teaching of statistics. Discourse on classifying
pedagogical actions into appropriate categories depended on the negotiations that took
place between the researcher and the teachers, and was bound to differ from one
teacher to another. The interpretations of the lessons and post-teaching discussions
could be viewed as temporal (dependent on time and pairs) and tentative. The possible
significant errors could be minimised by using multiple strategies to collect data.
•
Another limitation included external validity or the ability to generalise the results.
Only four teachers participated in the entire study. The number of cases was limited to
making broad generalisations. Not only the number of cases, but also the geographical
location and the school types may be too limited to produce a general theory on PCK
appropriate for teaching statistics in school mathematics. The number of participants
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also provided the possibility of variation or similarity in PCK assessment for
mathematics teachers using the same working document such as the mathematics
work schedule, the results in the senior certificate examination in mathematics,
recommendations from principals, subject specialists and peers.
•
Organising lessons outside normal school hours posed its own challenges. Learners
were sometimes tired at the end of the school day. Extra-curricular activities at the
schools occasionally affected the teaching programme. Therefore adjustments had to
be made to assure consistency and uniformity in all the statistics topics.
6.8 The role of the researcher in the non-participatory lesson observation.
In this study, non-participatory classroom observation in statistics lessons was conducted
with the four participating teachers. As explained in paragraph 2 of section 6.7, assessment of
teaching competencies involving a non-participating observer is usually associated with
inherent limitations, owing to the presence of the observer. The teacher and the students
might behave differently from the ways in which they would normally comport themselves.
(Cresswell, 2008; University and College Union, 2012). In this study there was always the
possibility that the participating teachers teaching could have been somewhat influenced, as
indicated later, by the presence of the researcher and the research interest. The interest was
with determining how mathematics teachers considered competent developed their PCK in
statistics teaching by observing them in statistics lessons among other things. Any one of
such influences could possibly occur in the planning and presentation of statistics topic
lessons. For example, to perhaps try to impress the observer they could select instructional
materials and use instructional strategies they think are effective but not necessarily
economical that they would not normally use routinely in teaching the assigned statistics
lessons.
The presence of the researcher could also have influenced learners’ responses or active
participation (freely or inhibited) during the lessons. While these are the possibilities that
could have arisen during the lesson, I believe that I tried to minimise those instances by first
introducing myself to the participating teachers and their learners during negotiations with
them on the extent and nature of the lessons to be taught and observed (Creswell, 2008,
University and College Union, 2012) and spending some time with the teachers
(familiarisation) before embarking on any formal classroom observation. Additionally, during
189
the meetings at which the teachers were briefed about the objectives of the research, they
were assured that the observation was not an assessment in any form or shape of their
teaching performance, but was designed to gain better understanding of how to help teachers
with a new topic, statistics, that has recently been introduced into the mathematics
curriculum. The learners were also encouraged by both the researcher and their teachers to
feel free and less anxious to participate, just as in a normal lesson, since no assessment was
involved. The participating teachers were given access to all the recorded field notes, the
video recordings and their transcriptions to comment on, and to approve, before the analysis
of data. Furthermore, triangulation of data helped to minimise and/or address any
inconsistencies in the participating teachers’ questionnaire and interview responses, and
classroom behaviour.
6.9 Summary of the chapter
In this chapter, the summary, conclusion and recommendations for further investigations
were presented. The results of the study indicate that mathematics teachers may have
constructed PCK in teaching statistics through the acquisition of formal subject matter
knowledge of the topic in formal education programmes, and they develop their subject
matter content knowledge during classroom practice. The teachers taking part in the study
possessed the necessary content knowledge, and demonstrated it through procedural and
conceptual approaches to teaching statistical graphs, although the rule-oriented procedural
approach was dominant in teaching data-handling topics. Mathematics and statistics-related
textbooks and other learning materials were other sources used by the teachers to acquire the
subject matter content knowledge that was needed to plan and deliver their lessons.
Knowledge of instructional strategies, notably the use of a formal rule-oriented approach and
instructional skills such as the construction skills, was developed through formal education
and years of experience in classroom practice. Analyses of learners’ classwork, homework
and assignments were used mostly to gain teacher knowledge of learner misconceptions and
topic-specific learning difficulties. Intervention strategies such as the used extra tutoring,
class activities in the form of drill and practice, repeating and re-explaining of lessons in
which learners are experiencing difficulties as well as post teaching discussions were used to
addressed the alternative conceptions and learning difficulties. The chapter concluded with
190
highlights of the educational implications, suggestions and limitations of the study for future
researchers to note.
191
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205
APPENDIX
APPENDIX I
206
Kindly indicate your willingness to participate in the research on voluntary basis by signing
the space provided below.
Yours sincerely,
IJEH, SUNDAY B. (Researcher).
I, Mr/Mrs/Miss___________________________of____________________________ high
school have agreed to participate in the research project in mathematics education conducted
by Mr Ijeh, Sunday B.
___________________________________
___________________________
Signature of Participant
Date
207
APPENDIX II
I, /Mr/Mrs/Miss_____________________________ being the father/mother have agreed that
my child will attend lessons/ participate in the research project in Mathematics education
conducted by Mr. Ijeh, Sunday B (Researcher).
________________________________
___________________________
Signature of Parent
Date
208
APPENDIX IIIA
to teaching some aspects of statistics on scheduled dates.
I undertake to maintain confidentiality and that neither the school nor the mathematics
teacher involved in my research will be identified, and, will be free to withdraw at any time.
209
In line with the department regulations, a letter of consent will be given to the mathematics
teacher recommended by you, requesting for his/her voluntary participation.
I will be very appreciative of your assistance in this regard. Kindly indicate your willingness
by signing the space provided below.
Yours faithfully
IJEH, SUNDAY B.
(Researcher)
I, Mr/Mrs/Miss _________________________ the principal of ______________________
hereby grant Mr Ijeh, Sunday B. (Researcher) the permission to conduct a research project on
the topic indicated above.
______________________________
__________________________________
Signature of the Principal
Date
210
APPENDIX IIIB
211
212
APPENDIX IV
Criteria for Validating Interview Schedule for Teacher on How They Develop PCK in
Statistics Teaching.
Preamble
An educational background means where and what school you attended within a particular
period. Basically all the schools that one has been to study a given or a particular subject
(DoE, 2008). It comprises the university attended, courses/modules studied, qualification
obtained and duration of the study. According to Bucat (2004), subject matter content
knowledge is the knowledge of the subject matter about what should be taught and how it
should be taught for effective learning. The outstanding teacher is not simply a ‘teacher’, but
rather a ‘history teacher’, a ‘chemistry teacher’, or an ‘English teacher’. While in some sense
there are generic teaching skills, many of the pedagogical skills of the outstanding teacher are
content-specific. Beginning teachers need to learn not just ‘how to teach’, but rather ‘how to
teach mathematics’, how to teach world history’, or ‘how to teach fractions in mathematics.
With these skills, subject content knowledge can be transformed into pedagogical content
knowledge (PCK) (Geddis, 1993). In order to be able to transform subject matter content
knowledge into a form accessible to students, teachers need to know a multitude of particular
aspects about the content that are relevant to its teachability (Bucat, 2004). Those teaching
aspects that the teacher needs to know included the topics, method of teaching, effectiveness
of the lesson, nature of the topic, how to assess learners’ understanding of the topic and
effective participation, instructional strategies used and relevance of the topic to the learners.
Others included are how to identify learners’ learning difficulties and the intervention used to
address the learning difficulties such as workshops, extra tutoring and more problem solving
activities that can enhance learners’ participation in the topic or subject. Kindly indicate in
the space provided whether the attached interview covered what it supposes to cover in terms
of assessing the mathematics teachers’ content knowledge and educational background that
enabled them to develop their PCK in statistics.
213
1)
2
Educational Background and subject matter content
knowledge
a) Does the schedule request for the university/college
attended?
Options
b)
Does the schedule request for the participants’
qualifications?
Yes/No
c)
Does the schedule request for the
course/module/subject studied in the
university/college?
Yes/No
d)
Does the schedule request for how the module/subject
help in lesson preparations?
Yes/No
e)
Does the schedule request for how the teacher knows
that his teaching was effective?
Yes/No
f)
Does the schedule request if the teacher has interest in
teaching mathematics?
Yes/No
g)
Does the schedule request for how the teachers
understand the nature of the subject/topic?
Yes/No
h)
Does the schedule request if learners understand the
topic?
Yes/No
i)
Does the schedule request if the learners enjoy the
topic?
Yes/No
j)
Does the schedule request for how the teachers update
their content knowledge for teaching the topic/subject?
Yes/No
K)
If the teachers attend workshop for instance, does the
schedule request to know how effective was the
workshop?
Yes/No
l)
Does the schedule request to know if the facilitators of
the workshop are mathematics teachers or not?
Yes/No
m)
Does the schedule request for the duration of the
workshop?
Yes/No
n)
Does the schedule request for what was benefited from
the workshop?
Yes/No
o)
Does the schedule request if the workshop participants
need similar workshop in subsequent time?
Yes/No
Yes/No
Instructional skills and strategies
a)
Does the schedule request if the teachers are adhering
to the instructional approach as recommended in the
NCS curriculum?
Yes/No
b)
Does the schedule request for how learners can be
assisted if they experience some learning difficulties
based on the instructional approach used by the
teacher?
Yes/No
214
Response
3
c)
Does the schedule request for instructional skills and
strategies used for teaching statistics?7
Yes/No
c)
Does the schedule request for other instructional
approach used by teachers apart from the
recommended approach according to NCS?
Yes/No
d)
Does the schedule request for how learners learning
difficulties were resolved if any?
Yes/No
Learners’ learning difficulties
a)
Does the schedule request for the learning difficulties
which learners encounter during teaching?
Comments
________________________________________
NAME AND SIGNATURE OF RATTER
215
Yes/No
APPENDIX V
TRANSCRIPTION OF VIDEO RECORDS OF FIRST LESSON OBSERVATION OF
TEACHER A
The teacher came into the classroom and began the lesson as follows:
Teacher A:
Good afternoon learners?
Learners:
Learners answered, Good afternoon sir.
Teacher A:
“Let somebody tell me how to calculate mode, median and mean of
ungrouped data?”
Learner:
“Mode is the number that appears most often in a distribution. For
example; 1, 2,3,2,5. The mode is 2. For the median, the learner continued,
you arrange: 1, 2. 2. 3, 5. Therefore, Median is 2, because it is the middle
number after arranging the numbers according to size”.
Teacher A:
Wrote down the numbers mentioned by the learner as an example of the
data and requested another learner to tell him how to calculate mean. In
their mother tongue, he said, ke bokae? Many learners raised their hand to
answer the question but the teacher A nominated one of the learners to
calculate the mean with the data on the chalkboard.
Learner:
Mean is the average of the numbers. i. e. Mean =
Mean =
1+ 2 + 3 + 2 + 5
,
5
12
= 6.0
5
Teacher A:
Gave an example and explained to the learners how to prepare the
frequency table. How many members are their ages within 16-20?
Learner:
One of the learners counted and said: it is 7. Another learner said it is 6
(The correct one).
Teacher A
explains:
The class boundaries are calculated thus,
15 + 16
= 15.5,
2
20 + 21
16 + 20
= 20.5, etc; Mid-values =
=18; and fx is calculated as: 6
2
2
x 18 = 108.
Teacher A:
“I have completed the first three rows.” Then, “complete the remaining
rows by calculating the frequencies, class boundaries, mid-values and fx”.
Teacher A:
Is it clear? In their mother tongue, he said, le a nkutlwa?
Learner:
Yees Sir.
Learner:
Completed the frequency table individually. “I have completed mine” (The
learners who have finished raise their hands).
216
Teacher A:
Constructed the histogram by drawing the vertical and horizontal axes on
the chalkboard, choose scale, label the axes with class boundaries as on the
table and draw the bars on the axes (Teacher A drew three bars and asked
them to complete the remaining bars).
Learner:
Listened and watched how the teacher constructed the histogram using
topic specific construction skills (Drawing of axes, choosing of scale and
labelling of axes, drawing the line of best fit).
Teacher A:
“Now complete the histogram.”
Learner:
Completed the histogram individually.
Teacher A:
Went round to check how learners were constructing the histogram and
further to answer the follow-up questions (By explanation).
Teacher A:
Calculated the mode from a histogram by drawing a diagonal form the top
right corner of the highest bar of the histogram to the top right corner of
the next bar on the left hand side and draw a second diagonal from the top
left corner of the highest bar to the top left corner of the next bar on the
right of the highest bar. He further refers the learners to how mode is
calculated in a stem and leaf diagram and to use that method for
confirmation of the answer obtained.
Learner:
Learners did as teacher A explained with their graph sheet on individual
basis.
Teacher A:
Analysed and interpreted the histogram i.e. 7 members of a netball club are
within the ages of 16-20 years. 67% of the members of the club are within
the ages of 21-30 years (e.g. add all numbers within the ages 20 – 30
divide by 27 and multiply by 100).
Learner:
Watched how the teacher calculates the percentage of learners within the
ages of between 21-30 years and write it on their notebook.
Teacher A:
Now, do this as classwork.
Learner:
Did classwork by preparing class boundaries, drawing of axes, choose
scale for drawing and labelling the axes, draw the line of best fit (bars).
Teacher A:
Monitored and guided learners as they did the classwork.
Teacher A
commented:
“I can see that your (some of them) diagrams are not correct. Make sure
that you have chosen the correct scale as in the example, otherwise you
diagram cannot be correct. Some of you have constructed the histogram
very well, but many have not, because you choose a wrong scale. Please,
go back to your example and see how we choose the scale and do the same
for this exercise.”
217
Learner:
Continued with classwork and were still experiencing some difficulties
about the construction and interpretation of histogram especially
determining the mode from the histogram.
Teacher A
explains:
“The lesson is about to end. Please, those of you who have not completed
their classwork should do so at home and bring it to school tomorrow.
Here is your homework (referring them to the exercise on their
mathematics textbook) which you have to submit with the classwork you
could not complete. I want to see those of you who could not complete
your classwork immediately after closing tomorrow so that I can assist you
on those areas where you are experiencing problems.”
218
APPENDIX VI
RANSCRIPTION OF VIDEO RECORDS OF SECOND LESSON OBSERVATION
OF TEACHER A
The teacher came into the classroom and began the lesson as follows:
Teacher A:
“Good afternoon learners?”
Learner:
Learners answered, “Good afternoon sir”.
Teacher A:
“Let me see how you did the homework which I gave you yesterday?”
Learner:
Opened to the page in their mathematics notebooks where they did the
homework.
Teacher A:
Checked and marked the homework from one learner to the other.
Teacher A:
Solved homework on chalkboard.
Learner:
Wrote correction on notebook.
Teacher:
Provided a photocopied exercise on ogive and requested learners to
interpret it. That is; calculate first, 2nd, 3rd quartiles, minimum and maximum
values.
Learner:
Find first quartile, 2nd quartile, third quartile, minimum and maximum
value in Groups.
Teacher A:
Used the values got from the interpretation of graph to construct a box-and
whisker plot while learner watched.
Learner:
“Could you please explain again how to calculate the first and third
quintiles?”
Teacher A
Using the formula as in previous examples, Q1 =
Explains:
1
(N + 1)th position you
4
Can find the position of the first quartiles and Q3 = ¾ (n + 1)th can be used
to find the position of the third quartiles Q3 can be traced from the
cumulative frequency to the curve down to the horizontal axis to determine
the value of the first quartile. The same applies to the value of the third
219
quartiles. While Q1 = 52, Q2 = 63 and Q3 = 73 the maximum 100 (using a
similar example in their textbook for explanation).Further interpretation:
25% of the learners got less than 52%, 50% the learners got less than 63%
and 75% of the learners got less than 73%.
Teacher A:
“The formula can also be applied to ungrouped data. You may apply it to
the exercise you did previously in ungrouped data.”
Learner:
Wrote the reference for the homework and noted it in their textbooks as
indicated above.
Teacher A
“It appears that some of you do not understand how to calculate the
Comment:
quartiles in ungrouped and grouped data. Can I see you tomorrow at 15h00
to explain more?”
Learner:
“Thank you, see you tomorrow.”
220
APPENDIX VII
TRANSCRIPTION OF VIDEO RECORDS OF FIRST LESSON OBSERVATION OF
TEACHER B
Teacher B:
Greeted the learners, “Good afternoon class”
Learner:
“Good afternoon sir?”
Teacher B:
“Can we move to the science laboratory because we want to use electricity in
our lesson today?”
Learner:
Moved to the laboratory before the lesson began.
Teacher B
“I want you to solve the exercise on preparation of frequency table on the
Commented: photocopied paper within 5mins”
Learner:
Prepared a frequency table.
Teacher B
“A bar graph is a pictorial representation of statistical data in the form of
explained:
rectangle called bar. A bar graph is often needed to compare two or more
values that are taken under different conditions or over time.” The frequency
table prepared (pre-activity) by the learners was used further used to explain
and construct a single bar graph. Teacher B drew the vertical and horizontal
axes, label vertical axis as frequency and horizontal as scores. Draw the bar
for each score”, teacher B said.
Learner:
Watched as the teacher explains how to construct and interpret the bar graph
and write explanation in their notebook and asked “How do we know that the
test is easy or difficult using these scores?”
Teacher B
“If the number of learners that scored 7 and above was more than six, then
Explains:
the test was easy.” (Teacher B read out the number of persons who scored 7
and above as 3 x 1 = 3.) “This means that about 60% of the learners scored
between 7 and 10. But if seven learners scored between 1 and 3 (teacher B
read from the graph), and the highest score was 5, the test was difficult, as
70% of the learners scored below 4 marks, and the highest score was 5.”
221
Thus, with a bar graph, it is easy to visualise and interpret learners’
performance in a test. From Figure 4.5.2a, it is evident that the test was
within the level of the learners, as the learners’ marks are not too low, and if
the pass mark is 4 (40%), then only four of the learners failed.
Teacher B:
After explanations, he gave classwork on the construction and interpretation
of bar graph (referring to their textbooks).
Learner:
Did their classwork as instructed by the teacher on one to one basis.
Teacher B:
Monitored and guided learners while they were doing their classwork.
Learner:
Some learners did not consider the concept of spacing which resulted to
misconception.
Figure A7:
Learners constructed a histogram instead of bar graph.
Teacher B:
Indicated to some of them that their classwork was wrong
Learner:
Demanded clarity why their classwork was wrong.
Teacher B:
Explained and re-explained and gave extra activities for learners to solve in
class and at home on how to solve the problem in the afternoon next day.
222
APPENDIX VIII
TRANSCRIPTION OF VIDEO RECORDS OF SECOND LESSON OBSERVATION
OF TEACHER B
Teacher B:
Greeted the learners “Good afternoon learners?”
Learner:
“Good afternoon sir?”
Teacher B:
Wrote the topic on the chalk board
Learners:
Watched and listened.
Teacher B
“Mention two ways of presenting data.”
Learner:
Mentioned frequency table, bar graph, pie chart, histogram
Teacher B:
Referred the learners to the exercise in their textbook. Explained how to
prepare a cumulative frequency table with the first three rows and instructed
the learners to complete the preparation of the frequency table with the
remaining rows.
Learners:
Completed the frequency table.
Teacher B:
Wrote all the cumulative frequencies calculated to ensure that every body
agreed on common cumulative frequency table.
Teacher B:
“Draw the vertical and horizontal axes like this .Label the axes with the
vertical as cumulative frequency and the horizontal with the marks value.
Plot the point (10, 0); (20, 2); (30, 8) etc. Continue with the remaining
points,” the teacher said
Learners:
Learners plotted the remaining points e.g. (40, 15); (50, 29); (60, 49), (70,
84); 80,113), (50, 29) ;( 90,119); 100,120); in groups
Teacher B
“This is how to interpret the graph.”
Learners:
Listened and watched as teacher B interprets the graph (Using the follow up
questions).
Teacher B:
Interprets the ogive by determining the quartiles e.g. 1st, 2nd and 3rd quartiles
as 52, 63 and 73 respectively.
223
Teacher B:
“Now do this exercise as classwork but with class internal beginning from
20-30, 30-40, 40-50, 50-60, while he monitored and guided them.
Teacher B
“Most of you appear not to know how to label the horizontal axis with class
Commented: boundaries beginning from 0-10 , 10-20, 20-30, etc and beginning from 20-
30, 30-40 40-50 etc. Look at how you can do it (referring the learner to a
graph paper and showing how to mark out the value on the horizontal axis.”
Figure A8. Learners constructed a histogram instead of an ogive.
Learners:
Tried to do as the teacher instructed, yet some were still experiencing
problems.
Teacher B:
Re-explains how to construct and interpret the ogive.
Teacher B:
“Now do the exercise at 8.11, 8.12, and 8.13 in your textbook.” Teacher B
instructed them to finish and submit the extra activities before going home
(The activities were divided into two: one part as homework for those who
were not experiencing problems and the other part for learners who were
experiencing some difficulties.
Teacher B:
Asked oral questioning as a way of concluding the lesson: ‘What does ‘n’
represent in the formula for calculating the quartiles? Where can I locate the
quartiles using the formula?’ Learners nominated by Teacher B gave
satisfactory answers which were followed by homework and post-teaching
discussion.
224
APPENDIX IX
TRANSCRIPTION OF VIDEO RECORDS OF FIRST LESSON OBSERVATION OF
TEACHER C
Teacher C:
“Good after noon learners?”
Learners:
“Good afternoon sir.”
Teacher C:
Specified the outcome of the lesson and said, “We are going to learn how to
construct, analyse and interpret the ogive. Before we do that, let us look at
the homework on the histogram.”
Teacher C:
Marked and checked the homework on histogram on a desk to desk basis.
Learner:
Some of the learners wrote corrections from friends who got the homework
correct for the questions they got wrong before the teacher could do that.
Teacher C:
“What is the difference between a class interval and class boundary?”
Learner:
“They are the same. Both of them contain groups of numbers.”
Teacher C:
“Mention various ways of representing data?”
Learners:
Mentioned bar chart, pie chart, scatter plots, line graph, etc.
Teacher C
“Let us prepare a cumulative frequency table using the table on the paper
Comment:
that I gave you.”
Teacher C:
Prepared the cumulative frequency of the first 3 rows and instructs learners
to complete the cumulative frequency table of the remaining rows.
Learner:
Completed the cumulative frequency of the remaining row.
Teacher C
“Now we can construct the ogive. Draw the two axes, label them using the
Explains:
Cumulative frequencies for the vertical axis and profit for the horizontal
axes. Now I will plot three points and you will plot and connect the
remaining points and lines of best fit for the curve.”
225
Learner:
Completed the plotting and connect the curve.
Teacher C:
Walked around the class from desk to desk, analysing learners’ classwork
and monitoring how learners are completing the plotting and connect the
curve.
Figure 4.5.3: An ogive showing the ages of cars of a sample of 100 car
owners.
Learner:
Continued with the completing of the ogive.
Teacher C:
“How do we calculate the median?”
Learner:
Teacher C:
Learner:
Quoted a formula; Q₁ =
1
(N + 1).
2
How do we calculate First quartile, second quartile and third quartile?
Quoted some formulae: Q1 =
1
(N + 1) and Q₃ = ¾ (N + 1). These formulae
4
were used to calculate the quartiles’ position as a way of interpreting the
graph and further explain how we calculate the quartiles.
Learner:
Listened and watched as he described how the quartiles were obtained.
226
Learners:
Some learners drew a histogram instead of an ogive. This is a
misconception.
Teacher C:
“Now I observed that some of you constructed a histogram instead of ogive
the question says construct an ogive and not a histogram”
Learner:
Reconstructed the ogive with the help of the teacher.
Teacher C:
Identified learners who are experiencing difficulties due to the
misconception and requested them to see him after the lesson one by one in
order to help them correct the difficulties they had.
Teacher C:
Summarises the lesson with oral questioning (How do you calculate the
cumulative frequencies? In constructing the ogive, and do you plot the
cumulative frequencies against the lower or upper class boundaries? He gave
the learners homework by referring them to their textbooks and other
statistics related materials.
Some of the learners who got their classwork correct went home at the end
of the lesson but those who were experiencing difficulties had to wait and
see the teacher one after the other for immediate assistance.
Learner:
Noted the homework given to them.
227
APPENDIX X
TRANSCRIPTION OF VIDEO RECORDS OF SECOND LESSON OBSERVATION
OF TEACHER C
Teacher C:
Indicate the Outcomes of the Lesson: The purpose of the lesson is to
learn how to construct, analyse and interpret scatter plots.
Teacher C:
Wrote the topic on the chalk board and introduces the lesson by giving
the learners some photocopied exercise to analyse as a pre- test
(requesting them to indicate how they have constructed the scatter plots
(analysis).
Learner:
Analysed the scatter plot in groups of two and three.
Teacher C:
Wrote a table and requested for a volunteer to plot the point on the
chalkboard as a way of explaining more about the construction of
scatter plots.
Teacher:
Teacher C walked around the class to monitor how learners are
analysing the scatter plots. He further asks, “Do you need to draw a line
of best fit in order to determine how the variable x and y are
connected?”
Learners:
“Yes sir.”
Teacher:
“Some of you interpreted diagram C as having negative correlation.
Why?”
Learner:
“Because the points are scattered all over (misconception).”
Teacher C
“No; only one point stood out of others as outliers. It has little or no
Explains:
impact on the correlation of the two variables. He presents more
photocopies of related examples in real life situation (see table 4.4.4c).
Learners:
“How do you account for outliers in a scatter plot?”
228
Teacher C:
“Outliers of two or more can affect the correlation of two variables and it
depends on the number of the correlating points.”
Learner:
Wrote explanations.
Teacher C:
“Why do we say that diagram A has a strong positive correlation?”
Learner:
Explains how the points are clustered along a given line indicating that
the more learners are taught, the more they perform in the test. This
shows a relationship between learner performance and the period they
were taught.
Teacher C:
Summarises the lesson with oral question and gave homework.
229
APPENDIX XI
TRANSCRIPTION OF VIDEO RECORDS OF FIRST LESSON OBSERVATION OF
TEACHER D
Teacher D:
Wrote the topic on the chalk board and gave learners photocopies of
statistics exercises. “Do the exercise I have given to you for 5mins”
Learners:
Solved the exercise individually involving the preparation of the
frequency table of given data.
Teacher D
explains:
After all learners had agreed on a common answer to frequency table
prepare, he showed the learners how to construct and interpret a bar
graph: “Draw the axes, label them with number of rows on the vertical
axis and accompanied on the horizontal axis constructing the bar
graph.”
Learner:
Listened and watched the teacher as he constructed the bar graph. In
their mother tongue, he said, labella ga ke go bontsha.
Teacher D:
After he had finished, he asked the learners, “What was the first thing I
did, when constructing a bar graph?”
Learner:
“You draw the vertical and the horizontal axis and label it”
Teacher D:
Which company manufactures the least number of cars?
Learner:
“Tata”
Teacher D:
“Why do I have to leave space between the bars?”
Learner:
“To show that they are different companies.”
Teacher D:
Gave a classwork on construction and interpretation of the bar graph.
Learner:
“Do the classwork by constructing and interpreting the bar graph and
find out how many learners fail the test if the pass mark is 5.”
230
Teacher D:
Monitored and guided learners as they were doing their classwork.
Learner:
“Why do we need to leave space between the bars?”
Teacher D:
“All companies manufacture cars but of different types (or different
companies)”
Learner:
Some learners drew a histogram instead of a bar graph by not leaving a
space between the bars. Some did not consider the constancy of equal
spacing between the bars.
Teacher D
Re -explained the constructed bar graph as some learners are still
explained:
experiencing difficulties in terms of constructing the bar graph as
explained above. And a bar graph is not a histogram as some of you
have done. While a bar graph have a common space between the bars,
a histogram does not.
Learner:
Listened, watched and wrote explanation on their notebooks. In their
mother tongue, he said, labella ga ke go bontsha.
Teacher D:
Gave homework on construction and interpretation of bar graphs.
Learner:
Wrote down the homework.
231
APPENDIX XII
TRANSCRIPTION OF VIDEO RECORDS OF SECOND LESSON OBSERVATION
OF TEACHER D
Teacher D:
Requested for the homework given to the learners in the previous lesson.
Learner:
Presented their completed homework individually.
Teacher D:
Checked and marked the learners’ completed homework on stem and leaves.
Learner:
Wrote corrections on the homework (for those who got some answers wrong).
Teacher D:
Wrote the topic on the chalkboard (construction, analysis and interpretation of
histogram).
Learner:
Listened and watched.
Teacher D:
Presented photocopies of exercise which was used to explain the topic.
Learner:
Received the photocopy and watched as the teacher demonstrated how to
construct, analyse and interpret a histogram.
Teacher D:
Prepared the frequency table of the data (To determine
preconception).
Learner:
Prepared the frequency table using a given class interval.
Teacher D explained: “This is how you draw a histogram. Draw the vertical and horizontal
axis. Label the vertical as frequency and horizontal axis as masses of
the player. Join the line of best fit in the form of a rectangle. I will draw two
rectangles and you will complete the remaining one.”
Learner:
Completed the histogram as the teacher had instructed.
Teacher D:
Interpreted the histogram by determining the measures of the central tendency
(Mode, Mean) that best describes the players according to their weight, and
gave learners classwork.
Learner:
“Why is it necessary to start marking the horizontal axes with 70 and not 0?”
232
Teacher D:
“You may make a zig-zag to indicate that you did not start from 0 or start from
the vertical line as shown in the table of values. You will have enough space to
construct the histogram.”
Learner:
Looked on for some time as a way of showing that they were not satisfied with
the explanation. They noted it and used the same method to do their class
work.
Teacher D:
Summarised and concluded the lesson with more explanation on the examples
and gave them homework on the same topic. Some of the learners who were
still experiencing some difficulties about how to construct a histogram of
grouped data were given extra compulsory activities to solve after the lesson
from their recommended textbook. “All of you who failed this activity have to
do this exercise and see me tomorrow after the normal school hours or closing
so that I can explain to you more about how to construct histogram.”
233
APPENDIX XIII
Criteria for validating questionnaire schedule for teachers on how they develop PCK in
statistics teaching.
Preamble
The attached questionnaire aims at investigating what the teachers actually did while teaching
such as the method applied, content of the lessons, nature of the topic, how the teacher
identified the learners preconceptions and learning difficulties, how the difficulties were
resolved if any and how the lessons were evaluated. Kindly indicate with the options
provided, your opinion about using the schedule to assess what the teacher actually did while
he was teaching statistical graph during the case study period.
S/No
Descriptions
Option Respond
1
Instructional strategies used for teaching statistical graphs
A
Does the questionnaire asked for the duration of the lesson?
Yes/no
B
Does the questionnaire request for the topic of the lesson?
Yes/no
C
Does the questionnaire request for the objective of the lesson?
Yes/no
D
Does the questionnaire request for the prior knowledge the
lesson needed?
Yes/no
E
Does the questionnaire request if learners have prior knowledge
of the topic?
Yes/no
f
Does the questionnaire request for how the teacher identifies the
preconception with which learners come to the class about the
topic?
Yes/no
G
Does the questionnaire request whether learners achieved the
objective of the lesson or not?
Yes/no
H
Does the questionnaire request for how learners responded to
class activities, homework and assignments?
Yes/no
I
Does the questionnaire request if the teachers were able to
follow the planned lesson from beginning to the end?
Yes/no
J
Does the questionnaire request for how teachers will improve
their lesson if their lesson was not successful?
Yes/no
K
Does the questionnaire request whether teachers evaluate their
lesson or not?
Yes/no
L
Does the questionnaire request how the teachers evaluate their
lessons?
Yes/no
M
Does the questionnaire request the reason for evaluating a
lesson?
Yes/no
234
S/No
Descriptions
2
Learning difficulties in the teaching of statistical graph
a
Does the questionnaire request for information about learning
difficulties that learners are experiencing?
b
Does the questionnaire request for how teachers resolve
learners’ learning difficulties if any?
c
Does the questionnaire request for what makes the learning of
statistics easy or difficult?
235
Option Respond
APPENDIX XIV
Criteria for validating written reports schedule for teacher on how they develop PCK in
statistics teaching
The attached is a teacher written report schedule for a period of four weeks for teaching
statistical graph. The schedule focuses on what has made the lessons easy or difficult as well
as where the learners’ learning difficulties lie during the teaching of statistical graphs for a
period of 4 weeks. Kindly indicate with the options provided, your opinion about using the
schedule to assess what has made the lesson easy or difficult.
S/No
Descriptions
Option Response
1
Learners’ learning difficulties
A
Does the written report schedule request for information about
the learning difficulties the teacher identifies when teaching the
statistics?
Yes/no
b
Does the written report schedule request for the difficulties that
the teacher experiences when teaching statistical graphs?
Yes/no
C
Does the written report schedule request for what the teacher
finds interesting or difficult when teaching the statistics?
Yes/no
D
Does the written report schedule request why the teacher finds
certain topic interesting when teaching?
Yes/no
E
Does the written report schedule request what the teacher find
less difficult to teach in the topic?
Yes/no
F
Does the written report schedule request for how the teachers
identify the preconceptions and misconceptions which learners
have about statistics during teaching?
Yes/no
G
Does the written report schedule request the preconception
identified when teaching statistics?
Yes/no
H
Does the written report schedule request for the misconceptions
identified by the teacher when teaching the topic?
Yes/no
J
Does the written report schedule request for how the teachers
address the misconceptions which they identified when
teaching the topic.
Yes/no
2
Instructional skills and strategies used for teaching
a
Does the written report schedule request for how learners
respond to class activities, homework and assignments
b
Does the written report schedule request for the changes that
the teacher will make next time with regards to the difficulties
encountered while teaching the topic both on the part of the
teacher or the learners’.
236
APPENDIX XV
Criteria for validating document analysis schedule for teachers on how they develop
PCK in statistics teaching
The teacher and learners’ portfolios, learners’ workbook and recommended mathematics
textbooks are the documents that will be used to examine if mathematics teachers are
complying with the National Curriculum Statements (NCS) policy for teaching and learning
of mathematics and have sufficient content knowledge of school statistics. Using these
criteria listed in the table below, kindly indicate with the options provided if the documents
contain adequate information that can be used to determine how well the mathematics
teachers are complying with implementation plan according to NCS.
S/No
Descriptions
Option
1
Learners’ workbook
a
Authenticity
Yes/No
i
Is learners’ workbook a genuine instrument for capturing
where their learning difficulties lie?
Yes/No
b
Credibility
Yes/No
ii
Can the workbook be used to gather enough evidence that
is free from error and distortion about learners’ learning
difficulties in statistics teaching?
c
Representativeness
iii
Can the evidence obtained from the learners’ workbook
give a true representation about the learning difficulties
they have in statistics teaching.
d
Meaning.
iv
Is the evidence about the learners’ learning difficulties,
gathered with the learners’ workbook, clear and
comprehensible?
Yes/No
2
Learners’ portfolio
Yes/No
a
Authenticity
Yes/No
i
Is learners’ portfolio a genuine instrument for capturing
where their learning difficulties lie?
Yes/No
237
Yes/No
Response
S/No
Descriptions
Option
b
Credibility
Yes/No
ii
Can the Learners’ portfolios be used to gather enough
evidence that is free from error and distortion about
learners’ learning difficulties in statistics teaching?
Yes/No
c
Representativeness
Yes/No
iii
Can the evidence obtained from the learners’ portfolios
give a true representation about the learning difficulties
they have in statistics teaching.
Yes/No
d
Meaning.
iv
Is the evidence about the learners’ learning difficulties
gathered with the learners’ portfolio clear and
comprehensible?
Yes/No
3
Teachers’ portfolio
Yes/No
a
Authenticity
Yes/No
i
Can the teachers’ portfolio be used to gather genuine
evidence about how they (teachers) are complying with
the teaching and learning policy in mathematics such as
using the work schedule, instructional strategies used for
teaching and learning difficulties that learners
encountered?
Yes/No
b
Credibility
Yes/No
ii
Can the teachers’ portfolios be used to gather enough
evidence that is free from error and distortion about work
schedules, instructional strategies used for teaching and
the learning difficulties encountered by the learners?
Yes/No
c
Representativeness
iii
Can the evidence obtained from the teachers’ portfolios
give a true representation about the learners’ learning
difficulties in statistics teaching?
d
Meaning.
iv
Is the evidence about the learners’ learning difficulties
gathered with the teachers’ portfolio clear and
comprehensible?
4
Textbooks
a
Authenticity
i
Is/are the textbook(s) the recommended textbook(s) for
teaching and learning in mathematics in the school?
Yes/No
Yes/No
238
Yes/No
Response
S/No
Descriptions
Option
b
Credibility
ii
Is the textbook(s) used error free in terms of the content of
statistics in school mathematics
Yes/No
c
Representativeness
iii
Does the textbook(s) contain adequate statistics content in
school mathematics according to the National curriculum
statements (NCS).
d
Meaning
iv
Is the content of statistics in school mathematics in the
textbook(s) clear and comprehensible?
239
Yes/No
Yes/No
Response
APPENDIX XVI
Criteria for validating the lesson plan and observation schedule.
The attached lesson plan/observation schedule was adopted from the Department of
education for classroom practice (DoE, 2009). Please indicate with the options provided, if
the schedule contains enough information for assessing a normal classroom practice in terms
of lesson planning/observation what the teacher did while teaching an assigned topic.
S/No
Description
Option
1
PLANNING
a
Does the schedule request for lesson topic?
Yes/No
b
Does the scheduled request for learning outcomes?
Yes/No
c
Does the Schedule request for assessment standard?
Yes/No
d
Does the schedule request for resources used during the Yes/No
lesson?
2
Pedagogical issues
a
Does the schedule request for how the lesson was Yes/No
introduction?
b
Does the schedule request for general handling of the class
e.g.
i) Classroom organisation?
Yes/No
ii) Discipline?
Yes/No
iii) Classroom interaction?
Yes/No
iv) Movement?
Yes/No
v) Learning climate?
Yes/No
vi) Involvement of the learners?
Yes/No
c
Does the schedule
(progression)?
request
for
d
Does the schedule request for how lesson is consolidated?
e
Does the schedule request for the description of the lesson in
terms of:
240
lesson
development Yes/No
Yes/No
Response
i) Language?
Yes/No
ii) Questioning techniques?
Yes/No
iii) Assessment?
Yes/No
iv) The use of resources?
Yes/No
v) Knowledge of the teacher?
Yes/No
vi) Errors and misconceptions
Yes/No
3
Learner related activities
a
Does the schedule request learners’ related activities?
4
Teacher related activities
a
Does the schedule request teacher related activities?
5
Evaluation/Conclusion
a
Does the schedule request how a lesson is concluded?
Comments
_____________________________________
NAME AND SIGNATURE OF RATTER
241
Yes/No
Yes/No
Yes/No
Appendix XVII
Analysis of participants’ responses to interview, questionnaire and teachers’ written report
Table 4.7.1: Teachers’ responses to interview about teachers’ subject matter content knowledge in
statistics teaching
Items
Interview Question
Responses
Teacher A
Teacher B
Teacher C
Teacher D
Coding
1
Which university
/college did you attend?
Unisa &
University of
North-West
University of
Zimbabwe
Aerica
University,
Zimbabwe
Vista University,
South Africa
-all attended
university
2
What qualification did
you obtain?
BEd Maths Ed.,
BA Psychology,
& Dip. in Maths
Ed.
BSc Maths &
Statistics
BSc
Mathematics
BEd Maths Ed.,
SED Maths and
Biology
- 2 had degree in
maths education
-2 had B.Sc in
mats.
- one also had
diploma
3
What course,
subject/module did you
study at the
university/college?
Maths, Physical
Sc, & Ed
Psychology.
The importance,
advantages and
disadvantages of
different
instructional
strategies.
Maths, Statistics
& Edu.
Methods of
teaching,
advantages and
disadvantages
of different
strategies.
Maths & other
courses.
Different
instructional
strategies,
advantages and
disadvantages of
the strategies for
teaching various
topics..
Maths courses &
Maths method
course.
Advantages and
disadvantages of
different teaching
approaches.
-all 4 study mats
and education
courses.
Advantages and
disadvantages of
different teaching
strategies
4
How long did you study
this course/subject?
BEd is 3yrs &
Dip is 2yrs.
Four (4) years
Four (4) years
Two (2) & Four
(4) years
-One for 3yrs,
three of them for
4 yrs, and two 0f
them for 2 yrs in
Dip.
5a
If one of the courses in
(3) is mathematics
methodology, how did it
help you to prepare your
lessons for teaching?
Use of varied
instructional
strategies.
Varied formulae
and strategies
for teaching the
same topic
The courses
helped me to
prepare the
lessons using the
required format
and knowledge
for teaching with
the objectives of
the lessons in
mind.
It helps me for
planning in line
with the work
schedule,
assessment and
evaluation of my
lessons.
- Help to plan
lessons and vary
instructional
strategies.
5b
How do you know that
your teaching is
effective?
Response from
learners or
feedback to
class works,
assignments,
homework etc.
Response from
the learners to
classworks,
homework and
assignments.
Learner’s
response to class
activities,
homework and
examinations.
Feedback to class
activities,
homework and
other related
tasks on the
topic.
-Analysis of
learners’
responses to
classwork,
homework and
assignments.
6
Do you have interest in
the teaching of
mathematics? If yes / no
why?
Very well. I
love teaching.
Mathematics
helps one to
improve one’s
thinking skills
and provides
opportunity for
problem
solving.
Yes, because
answers are
always there for
a particular
question.
Yes. It is very
challenging but
interesting, as it
makes one to be
precise and
accurate.
Yes. It is straight
forward. It is
either you get it
right or wrong.
-All four have
interest in
teaching
statistics as
mathematics
improves ones
thinking skill and
opportunity for
problem solving.
242
7
What is your
understanding of the
nature of the statistics
you teach?
Statistics is
practical in
nature and an
aspect of
mathematics
that one can
apply in
everyday real
life situation
especially in
summarizing
data.
I am quite
comfortable
with the
concepts during
teaching
because of its
practical way of
solving
problems. It
helps to
organised and
summarized
data in a
meaningful
way.
It helps to
simplify complex
data into
understandable
one that can be
used for
interpretation and
analysis.
It is a practical
topic that allows
learners to
participate
actively
especially during
the construction
of statistical
graphs and
preparation of
frequency tables.
Statistics helps in
summarising
data in a
meaningful and
understandable
way.
-statistics help to
organised and
summarise data
in an
understandable
manner for
making
decisions. The
ways statistical
graph are
constructed make
it look practical
in nature.
8
Do learners understand
the topic?
Yes and I notice
it through their
response and I
make sure that
they understand
by employing
appropriate
method of
teaching which
the topic
demanded.
About 90% of
the learners
were very much
comfortable
with the topic.
Yes. Learners’
level of
participation
during the lesson
is high.
Yes, I understand
it through their
response to class
activities and
homework.
-all four claim
learners
understood their
lesson and it is
observed from
the way they
respond to
classwork,
homework and
assignments
9
Do learners enjoy the
topic if yes/no why?
Yes, through
their
involvement in
the lessons and
participation.
Yes. They really
like every
aspect and this
was evident in
the way the
learners
performed in
some of the
activities given
to them.
Yes. They
sometimes make
contributions and
explain with
confidence to
their classmates
what they do not
understand
during lessons.
Yes. Through
response to oral
questions, class
work and
interaction with
fellow learners.
- All four claim
that learners
enjoy their
lesson.
-The way they
enjoy the lesson
was noticeable in
the way they
interact with
their teachers and
classmate during
class discussion.
15
Have you attended a
mathematics workshop
or teacher development
programme?
Yes. Workshop
on NCS for
grades 10-12
mathematics,
Investec
Enrichment
Mathematics
Programme.
Yes. Workshop
was on NCS for
Mathematics
grades 10-12
Yes. I attended
several
workshops on
mathematics
especially data
handling which
lasted for a week,
3 days, etc.
Yes. I attended
many workshops
on teachers’
development in
content
knowledge
especially in data
handling. The
duration of the
workshop was 7
days.
-all four claim
they have
attended
workshop on
professional
development
programme in
maths and data
handling in
particular.
16
If your answer in (15) is
yes, what was the
content of the
workshop?
Teaching and
learning of NCS
mathematics
and methods for
teaching
How to
calculate
measures of
central tendency
and spread.
The workshop
was on NCS
mathematics.
The workshop
was on the new
topics in
mathematics and
other challenging
topics.
-The workshops
were based on:
- methods of
teaching NCS.
-Data handling.
New and
challenging
topics in the new
curriculum.
17
What was the duration
of the workshop?
Every Saturday
and during
school holidays
for one year.
7 hours
workshop on
the topic
indicated above.
243
6 hours
workshop on the
above mentioned
topic.
08h00 to 14h00
for four weeks
(every Saturday)
on data handling.
Durations for the
workshops were
6hrs, 7hrs and
8hrs per day for
four weeks and
8hrs per day
during holidays.
18
Were the workshop
facilitators mathematics
teachers or mathematics
experts?
Yes.
Mathematics
experts from the
universities.
Mathematics
teachers.
Yes.
Mathematics
educators from
the university
and colleges.
Mathematics
educators and
experts from the
districts.
-The workshop
facilitators were
mathematics
experts from the
university,
department of
education and inservice teachers.
19
As a mathematics
teacher, did you benefit
from the workshop?
Yes. Very well.
How to teach
some
challenging
topics in
mathematics
such as data
handling.
Did not learn
new concepts
but some
changes in the
curriculum like
data handling
that is new.
All about
statistics
especially lower
and upper
quartiles ranges
etc.
Not as much, I
was taught what
I already know.
-All four claim
that they benefit
from the
workshops as
they gain more
confidence in
teaching, became
abreast with
contemporary
issues in
mathematics
education.
20
Would you recommend
that similar workshops
be held for teachers?
Yes, because the
benefits are
enormous as it
gives
confidence in
my class
practice.
Yes. To provide
educators with
contemporary
issues with
regards to
teaching and
learning but not
teaching
educators like
learners.
Yes. Any time
that may be
convenient.
Yes. It helps to
refresh and
reflect on what is
already known
and be aware of
the contemporary
issues in the
teaching and
learning of
mathematics.
-all four claim
that they will
recommend for a
similar workshop
that can help
them to reflect
and refreshed
their knowledge
on the subject.
244
APPENDIX XVIII
Table 4.7.2:
Items
10
Participants’ responses to the interview, questionnaire and written reports about
teachers’ knowledge of instructional skills and strategies for teaching statistics
Interview Question
Responses
Teacher A
Teacher B
Teacher C
Teacher D
Coding
In your own opinion
The topic is
It looks
It is very
It is a lively
-Statistics help to
and based on your
good to be
understandable
important to be
topic and very
oganise data in
experience in the
integrated into
and helps in
integrated into
practical in
meaningful way
teaching of statistics,
mathematics. It
understanding
mathematics
nature. It can
to the users-
how do you see the
lends support to
other topics too
because of its
also help to
-it lends support
topic (statistics) in
other areas of
because of the
uses in everyday
understand other
to understand
mathematics?
mathematics
way it helps in
life and other
subjects.
other subjects.
because it is
organising
professions in
-The way data is
very practical in
information in a
terms of making
represented
nature and good
meaningful way.
information to be
makes it look
in summarising
understandable to
practical
information e.g.
the users.
especially
frequency table
11
statistical graphs.
Do your learners
Yes, the
Yes. OBE
Yes, they do and
Yes they do and
-All four claim
understand your
approach is
approach is the
I ensure that they
I encourage
that learners
lessons based on the
OBE learner
general teaching
participate
them to
understood their
instructional approach
centred for
method. But one
actively or be
participate
lessons based on
for teaching as
teaching
can change
actively involved
actively in the
the instructional
recommended in the
mathematics but
depending on
in the lessons.
lesson.
strategies used in
curriculum?
not all topics
feedback from
teaching.
required it.
learners.
-OBE is used.
Learners need
-Learners must
the basic
be actively
background
involved.
because the
topic is new.
12
If learners have
I will try to
I will conduct
I have to involve
I have to change
-Explanation
problems in
explain the topic
remedial lessons
them in the
my methods of
with varieties of
understanding the topic
with familiar
with the learners
discussion after
teaching, repeat
example,
based on the
examples and
concerned after
the lesson and
the lesson and
remedial lessons,
instructional approach,
situation as well
the normal
provide more
organise extra
problem solving,
what do you do to help
as solving more
school hours
class activities
lesson in the
and monitoring
them understand?
problems on that
with familiar
and ensure that
topic in which
strategies were
topic. I will try
examples related
they understand
they are
used to resolve
to organised
to real life.
the lessons by
experiencing
any difficulty
extra-tutoring
monitoring how
some difficulties
that may arose
for them after
well they are
to help the
during teaching
school hours.
doing the
learners.
and based of the
activities.
instructional
approach.
245
13
What other
Questioning and
Using teaching
Definition of
Explanation of
-Demonstration
instructional strategies
answers and
aids because it
basic terms,
basic concepts
method,
do you use for teaching
demonstration
enhanced
explanations and
with examples
questioning and
and why?
methods. The
understanding
using outcome
and more
answer, use of
reason for using
and encourages
based approach
problem solving
teaching aids,
these methods is
the development
(OBE) for
activities in real
organizing of
that the teaching
of manipulative
teaching. Extra
life situations.
extra tutoring
of statistics is
skills such as
tutoring is also
and definition of
very practical
graph
important in
basic terms with
and most
construction.
order to improve
examples were
material used
learners’
other
are within the
understanding of
instructional
environment of
the topic and
strategies used
the learner (real
enhance their
for teaching
life)
participation in
statistics.
the study of
-The reason for
statistics.
using these
strategies is to
improve learner
understanding
and achievement
in the topic.
Items
1
Questionnaire
Responses
Teacher A
Teacher B
Teacher C
Teacher D
How long was the
Duration of the
Duration of the
45 minutes.
45 minutes.
lesson?
lesson was 45
lesson was 40
taught for 45
minutes.
minutes.
minutes each.
-Three teachers
-only one teacher
taught for 40
minutes.
2
Drawing and
-All four taught
representation by
interpreting of
statistical graph
statistical graphs
an ogive.
comprising of
What was the topic of
Statistical graph
Data
the lesson?
(histogram).
Scatter diagrams.
(communicative
histogram, ogive
frequency and
and scatter plots
ogives).
246
3
What was the objective
By the end of
By the end of the
By the end of the
By the end of
- All four claims
/ assessment standard
the lesson, each
lesson, each
lesson, each
the lesson, each
that the objective
of your lesson?
learner should
learner was
learner should be
learner should
of their lessons
be able to
expected to be
able to construct,
be able to draw,
was that learners
construct,
able to draw
analyse and
analyse and
should, at the end
analyse and
ogives and
interpret
interpret
of the lesson, be
interpret a
answer questions
statistical graph
statistical graphs
able to construct,
statistical graph
related to
such as the bar
e.g. cumulative
analyse, interpret
(histogram) as
constructing
graphs, ogive and
frequency curve.
and apply the
well as using
analysing and
scatter diagrams.
them to solve
interpreting of
knowledge to
real life
ogive using the
solve everyday
problems.
knowledge of
real life
measures of
problems.
central tendency
-
statistical
.
7
Did you think that the
Yes. Through
They did because
They did and I
Yes. Based on
-All four
learners achieved the
their active
they were able to
confirm through
their response to
participants
objective of the lesson?
participation in
respond
their response to
class activities.
claimed that the
the lessons and
positively to the
classwork and
objectives of
the reaction to
questions asked
homework.
their lessons
the assessment
by the teacher
were achieve and
task given
orally, in
it was evidence
afterwards.
classwork, home
in their responses
work and
to classwork,
assignments.
class discussion,
homework and
assignments
8
How did the learners
Positively and
The response
Excellent. Their
They performed
-All our teachers
respond to the class
they showed
was quite
responses were
well in the
reported that
activities, homework
interests in the
positive as the
good to indicate
activities given
learners’
and assignments?
topic.
learners wrote
that they
to them as
responses to
the homework
understood the
homework or
classwork,
activities.
lesson.
assignments.
homework and
assignments were
positive.
9
Were you able to
Yes and it was
Yes, the lesson
Yes and it was
Yes and
- All four
follow the lesson as
done as outline
was a successful
based on the
according to the
participants
planned at the end of
in the lesson
one from the
lesson plan.
lesson plan.
claimed that they
the lesson?
plans.
beginning to the
were able to
end.
follow the
planned lessons
from the starting
point to the end
247
12
How would you
By ensuring
By using more
By giving extra-
By giving more
-To ensure that
improve / sustain the
(through oral
teaching aids like
class activities
examples on the
learners has
lesson?
questioning or
the charts,
related to real life
topic and
adequate
pre-test) that the
overhead
and lessons.
solving past
background
necessary
projector etc.
questions.
about the topic.
background
-Explain lesson
knowledge does
with enough
exists before
teaching aids.
introducing new
_give enough
content.
and variety of
class activities.
-Solve past
questions on the
topics.
13
Do you normally
Yes (using class
Quite often.
Yes. Through
Yes, I do. I give
-All four
evaluate your teaching?
work, test,
Through class
class work, oral
classwork,
participants
exams,
work, test, exams
questions,
homework and
claimed that they
homework and
and homework.
homework.
sometimes
usually evaluate
assignments.
their teaching
assignments).
using classwork,
test and
examinations.
15
16
How do you evaluate
By asking oral
By observing the
As mentioned in
I use same
-By oral
your teaching
questions,
difference in my
question 13.
method as in
questioning,
performance?
during lessons,
teaching
question 13 to
classwork,
and giving
performance
determine
homework,
classwork,
through end of
performance.
assignments and
homework and
year
examinations.
assignments.
examination.
For what reason do you
To ensure that
To improve and
For personal
To evaluate my
-To ensure that
evaluate your teaching?
learners
adjust to the
professional
teaching
learners
understand what
performance of
development and
performance
understand.
they need to
the learners.
determine how to
based on
-To determine
know and to
give learners the
learners’
progress.
determine which
best of my
performance in
-To select a
methods of
capabilities.
classworks and
better method of
teaching may be
homework
teaching the
more effective
which
topic.
than the other
consequently
through
determines their
learners’
progress.
response to class
work and
homework.
248
Items
5
Teachers’ Written
Reports
Responses
Teacher A
Teacher B
Teacher C
Teacher D
How did learners
The learners
Learners were
Their responses
Many of the
-Learners’
respond to classroom
responded
able to write the
to class activities
learners did very
responses were
activities as well as
positively and
class activities as
were excellent.
well in their
positive,
homework or
showed much
well as
But few of them
classroom
excellent and
assignments?
interest in the
homework
need some help
activities as well
interesting. But a
classroom
efficiently with
to overcome their
as homework.
few of them have
activities. But a
exception of few
little difficulties.
Only few of
some difficulties.
few of them
learners..
them had some
problem.
have little
problem
6
What changes would
Try to use
I shall try to give
No change but
To adopt a
- Two of them
you make next with
different
learners more
more work can
different
use different
regards to the
teaching
work on the topic
be given to the
teaching
teaching
difficulties you
strategies or
to enhance their
learners for them
approach and
methods.
encountered while
approach and
understanding of
to understand
provide more
- Two of them
teaching, either on your
methods that
the topic.
more.
activities for
use more class
part or on the part of
will best suit the
them to solve.
activities for
the learners?
learners.
learners to solve.
249
APPENDIX XIX
Table 4.7.3: Participants responses to the questionnaire and written reports on teachers’ knowledge of
learners’ preconceptions and misconceptions in statistics teaching
Items
4
Questionnaire
Responses
Teacher A
Teacher B
Teacher C
Teacher D
Coding
What prior knowledge
Measures central
Calculation of
Measures of
Preparation of
-Measures of
does your lesson
tendency and
mean, mode,
central tendency
frequency table,
central tendency
require?
common bar
median, quartiles
and how to
class interval and
graphs.
and simple
interpret
boundaries, mid-
-How to prepare
additions and
information from
points in the case
frequency table
subtractions.
simple straight
of grouped data
from a given
line graphs.
representation.
data.
- Simple addition
and subtraction.
How to interpret
line graphs.
5
Do the learners have
Yes. Measure of
Yes. They could
Yes but not all
Yes. They have
-All four claimed
prior knowledge
central tendency.
even link the
especially about
been taught how
that learners have
(preconceptions) of the
previous
grouped data.
to construct and
prior knowledge
topic?
knowledge with
interpret line
of the topic they
the new concepts
graph and
teach.
such as
quadratic graphs.
histogram and
-Depending on
frequency
the topic.
polygon.
-As indicated in
question 4
6
Items
How did you identify
By asking
Through the
Asking refresher
By making them
- Using
the prior knowledge
diagnostic oral
correction of
questions
to participate in
diagnostic
(preconceptions)
questioning.
answers to
involving oral
solving some
techniques (oral
which the learners
previous
questioning.
short question at
questioning, pre-
came with to the class
questions
the beginning of
test
about statistical
(Homework)
the lessons; but
graphs?
that were given
sometimes you
Through analysis
to learners and
may ask oral
of learners’
pre-activities
questions.
responses to class
Teachers’ written
related to the
activities
topic I want to
(problem solving
teach.
and discussion)
Responses
reports
7
Teacher A
Teacher B
Teacher C
Teacher D
How do you identify
By continually
The
Through
Oral questioning
- Using
the preconceptions and
asking
preconceptions
responses to
at the beginning
diagnostic
250
misconceptions of the
diagnostic oral
can be identified
classwork and
of the lessons for
techniques e.g.
learners during
questions at the
through oral
homework.
preconceptions
Oral questioning,
teaching?
beginning of the
questioning or
and looking
pre-test,
lesson and
pre-test at the
Misconceptions
through their
classwork and
during the
beginning of the
were identify
responses to
homework.
lesson.
lessons.
through the
classwork and
Misconceptions
analysis of
homework on bar
were identify
were identify
Misconceptions
learners’
graphs in the case
through the
through the
can be identified
classwork and
of
analysis of
analysis of
by going through
homework on
misconceptions.
learners’
learners’
their classwork
scatter plots and
classwork and
classwork and
or homework.
ogive.
homework on
homework on
The
histogram.
misconception
Misconceptions
statistical graphs.
can be resolved
by learners in
more practice
exercise.
8
What preconceptions
Confusing the
Preconceptions:
They are familiar
The drawing of
- Preconceptions
and misconceptions do
concept such as
mode, median
with measures of
the bar graphs.
identified were
you identify?
mode, median
and mean.
central tendency
The
measure of
and mean. The
Misconception is
but develop some
misconception is
central tendency,
preconceptions
drawing a bar
misconceptions
that they tend to
construction of
identified are
graph instead of
such as the
draw a histogram
simple linear
mode, median
a histogram.
construction of
instead of bar
graphs and bar
and mean. The
bar graph instead
graph.
graphs.
misconceptions
of histogram
are inability to
during the lesson.
Misconceptions
draw graphs of
noticed were
grouped data,
construction and
choosing scale,
interpretation of
labelling axes
graphs of
and interpreting
grouped data e.g.
the graphs
histogram, ogive
especially for
and scatter plots
grouped data.
9
How would you
By dealing with
By providing
Give learners
Re-teach and re-
-Explanations
address the
clear and
varieties of
chance to
explain using a
with examples,
preconceptions and
practical
examples and
elaborate on their
different
more class
misconceptions, if any,
examples
activities for
misconceptions
approach and
activities and re-
learners to solve.
Identified during the
dealing with
in order to correct
give them more
teaching of the
teaching and learning
such concepts
the
activities on same
topic.
process?
until learners
misconceptions.
topic.
understand them.
251
APPENDIX XX
Table 4.7.4: Participants responses to the teachers’ interview, questionnaire and written reports about
teachers’ knowledge of learners’ learning difficulties
Items
14
Interview Question
What learning
Responses
Teacher A
Teacher B
Teacher C
Teacher D
Coding
Have problems
Construction of
Construction
Graphical
-learners’ learning
difficulties are
difficulties do you
with plotting of
graphs
and interpreting
constructions
remember
scatter plots,
especially
of graphs
and
inability to construct
experiencing as a
drawing of line
choosing an
especially
interpretation
graph of grouped data
pupil and as a
of best fit and
appropriate
ogives and
especially
e.g. ogive, histogram,
university / college
forming
scale in drawing
scatter
ogives, scatter
scatter plots and
student or from
equation from
histograms,
diagrams.
plots and
frequency polygon.
teaching experience in
the scatter plot.
frequency
regression lines.
polygons,
statistics?
ogives and
scatter plots.
Items
10
Questionnaire
Responses
Teacher A
Teacher B
Teacher C
Teacher D
What difficulties did
No major
Some learners
Not much
None except
- Inability to
the learner experience
problem and
could not
except graphs
determination of
construct graph of
during teaching?
but when it
choose an
construction and
mid points and
grouped data,
arises, I will
appropriate
interpretation of
construction of
calculate median, mid
deal with it
scale on the
grouped data.
graphs. But any
values and
Such as some
graph paper to
problem occur I
interpretations.
basic
construct graphs
will be able to
calculations
of grouped data.
deal with it
Anticipated learning
such as the
difficulty to be dealt
median and
with if they arose
mean and mode
of group data
11
How did you address
By giving them
I have to guide
By giving them
Provide more
-The difficulties
these difficulties?
more exercises
them on how to
more exercises
examples and
identify were
to solve on
choose a
to solve on
possibly repeat
resolved by given
problem areas.
suitable scale
problem areas.
the lessons
learners more
for a particular
activities to solve,
data hence we
guiding and
ended up using
monitoring of
the same scale
learners on how to
in constructing
construct graphs of
some of the
grouped data and
graphs for the
repeating the lesson.
sake of
uniformity.
252
14
What is it about
Measure of
The learning of
Example of
It is an
-learning of measures
statistics that makes
central
statistics was
everyday life
interesting
of central tendency is
the learning easy or
tendency is
quite easy
makes the
subject such that
simple to teach and
difficult?
easy because it
because the
lesson lively and
if one has a
learn.
is more
learners had the
interesting. The
deeper
practical to
background in
everyday life
understanding of
-When lessons are
teach.
grade 10
example makes
it, he/she will be
taught by an
Construction
especially for
learners to
able to present it
experience teachers.
and
measures of
easily
in a manner that
interpretation of
central tendency
understand the
learners can
-Graphs of grouped
graphs of group
except for the
lessons. But
understand it.
data are difficult to
data are
construction
graph of
However,
construct.
difficulty to
and
grouped data,
grouped data
teach.
interpretation of
especially
graphs are more
graphs of
scatter diagrams
difficult to
grouped data
are difficult to
construct and
such as
learn.
interpret.
histograms,
frequency
polygons and
cumulative
frequency
curves (ogives).
Items
Teachers’ Written
Responses
Reports
1
Teacher A
Teacher B
Teacher C
Teacher D
Coding
What learning
Basic
1. Choosing a
Some learners
Learners’
- Learning difficulties
difficulties did you
knowledge and
suitable scale.
are slow in
inability to
identify in learners
identify in learners
background
learning of the
interpret
are poor background
when teaching a
information
2. Neatness of
topic especially
statistical graphs
in the topic,
topic?
linking to the
axes and title of
the drawing of
such as the
construction of
new content.
the graphs. 3.
graphs of
histogram, the
grouped data
They were
Labelling of
grouped data. I
ogive and scatter
(labelling of axis),
identified by
axes and title of
observe this
plots. These
interpretation of
going through
the graphs of
while they are
difficulties were
graph of grouped data
grouped data.
doing their
discovered by
and choosing suitable
the learners’
classwork. I also
analysing
scales for
classwork and
All of them
use homework
learners’
constructing graph of
homework
were identified
and assignments
classwork,
grouped data. They
by analysis of
to discover their
homework and
were identified by
their classwork,
mistakes
assignments.
analysis of learners’
responses to
homework,
classwork,
assignments,
homework,
project, etc.
assignments, etc.
253
2
What difficulties do
The
As described in
None except
Non availability
-Construction and
you experience in the
construction of
(1) above.
that learners
of graph board
interpretation of
teaching of statistical
graphs of
struggle to
which make it to
graph of grouped
graphs?
grouped data.
construct graphs
inaccurately
data.
of grouped data.
display the
graphs during
teaching.
3
What do you find
The practical
The topic has
Frequency
Displaying
-The way the topic is
interesting in this
application of
some practical
graphs such as
graphs of the
relatable to everyday
topic and why?
statistical
examples that
the bar graphs
same data using
real life, graphical
concepts and
can be used in
because of easy
different graphs.
representation of data
knowledge in
day to day life
and smart
This reinforces
and how it
everyday life.
situations and
presentation.
the uses of
encourages
the topic can be
statistics in
manipulative skills.
used in solving
everyday life.
related
problems in
other subjects
such as research
projects.
4
What do you think
Mode, median
The topic is less
The stem-and-
Mode, median
-Mode, median and
you find less difficult
and mean of all
difficult to
leaf diagram,
and mean of
mean of ungrouped
to teach in the topic?
ungrouped data
teach and
mode, median,
ungrouped data
data.
learners enjoy it
mean and range.
and bar graphs.
especially
-Stem-and-leaf.
mode, median
and mean of
ungrouped data.
5
How did learners
The learners
Learners were
Their responses
Many of the
-Learners’ responses
respond to classroom
responded
able to write the
to class
learners did very
to classroom
activities as well as
positively and
class activities
activities were
well in their
activities such as
homework or
showed much
as well as
excellent.
classroom
classwork; homework
assignments?
interest in the
homework.
activities as well
and other related
as homework.
class activities were
classroom.
positive.
6
What changes would
Try to use a
I shall try to
No change but
To adopt a
- Changes expected
you make next with
different
give learners
more work can
different
of the teacher are
regards to the
approach and
more work on
be given to the
approach and
adopting different
difficulties you
methods that
the graph to
learners for
provide more
teaching approach,
encountered while
will best suit
enhance their
them to
activities for
providing more
teaching, either on
the learners.
understanding
understand
them to solve
activities for learners
of the topic.
more.
that may relate
to solve.
your part or on the
to real life.
part of the learners?
254
APPENDIX XXI
Table 4.8:
TEXTS
A Comparison of the documents used by participants in statistics teaching
CASE STUDY 1
CASE STUDY 2
CASE STUDY 3
Learner
The learner workbooks contain written The learner workbooks contain written and The learner workbooks contain
workbooks
and marked class work and home-
marked class work and home- work. For
work. For example, there was evidence example, there was evidence of learners’
of learners’ inability to construct and
inability to construct and interpret graphs
interpret graphs of grouped data due to of grouped data due to wrong scaling of
CASE STUDY 4
The learner workbooks contain
CODING
-Learners’ workbook contains written
written and marked class work and
written and marked class work and and marked classwork but the rate at
home- work. There was similar
homework. There was similar
evidence in case study 3 as in case
evidence in case study 4 as in case in each case.
which learners make progress differ
studies 1 and 2. Detail descriptions of studies 1, 2, and 3 but the rate at
wrong scaling of data axis (Learning
data axis such as histograms (learning
statistics concepts and mathematical
which learners made progress
difficulties) such as histograms,
difficulties). They were identified by
connections between them were
during the lesson was much better.
cumulative frequency curves, and the
analysis of learners’ classwork and
available in which quartiles were
There were details of definition of
drawing of lines of best fit in a scatter
homework. In some cases, learners did
described and used to interpret ogive. statistics concepts with examples
diagram. Learning difficulties were
well in terms of graphical constructions
Learners’ misconceptions (drawing a relatable to real life. Comparative
identified by analysis of learners’
and interpretations in bar (with horizontal histogram instead of bar graph) and
classwork and homework. Teacher A
and vertical bar graphs) and double bar
seem to teach with procedural
graphs, histograms and ogives. There was to label data axis due to wrong scaling misconceptions (drawing a
knowledge (construction of a
evidence of intervention to resolve
histogram) and conceptual (Defining
learning difficulties. Teacher B seemed to learners’ classwork and homework on vice versa) and learning difficulties difficulties of not being able to label
and explaining with examples the
be flexible in his approaches to the
graph of grouped data during
of not being able to label data axis data axis due to wrong scaling, and
meaning of mode, median, mean and
teaching of statistics. Learners’ were
marking. Scatter plots construction
due to wrong scaling were
misconception of drawing a histogram
histogram) approaches However, there sometimes given extra lessons.
done procedurally and interpretation
identified by analysis of learners’
instead of a bar graph and vice versa.
were also some instances where
examples and learners’ class activities classwork and homework of graph
learners performed well e.g.
relatable to real life was available in
of grouped data. Teacher D
construction, analysis and
the learners’ workbook. Teacher C
seemed to be flexible in his
interpretation of bar graphs, stem-and-
seemed to be flexible in his
approaches to the teaching of
leaf, histogram and ogives which they
approaches to the teaching of
statistics where he uses both
did individually and in groups
statistics. Learners’ were sometimes
procedural and conceptual
-Analysis of learners’ classwork,
homework and assignments were used
to identify learners’ misconceptions
and learning difficulties.
relationship between a bar graph
learning difficulties of not being able and a histogram. Learners’
were identified by analysis of
histogram instead of bar graph and -There was evidence of learning
knowledge to teach statistical
(flexibility in approaches). Detail
255
Teachers taught with both procedural
and conceptual knowledge in their
statistics lessons.
TEXTS
CASE STUDY 1
CASE STUDY 2
descriptions of concepts. etc Learners’
CASE STUDY 3
given extra lessons.
were sometimes given extra lessons.
CODING
graphs. Learners’ were sometimes
given extra lessons.
The learner portfolios contain written and The learner portfolios contain written The learner portfolios contain
Learner
The learner portfolios contain written
portfolios
and marked tasks such as assignments, marked tasks such as assignments, projects and marked tasks such as
projects and investigations, informal
CASE STUDY 4
and investigations, informal and formal
assignments, projects and
written and marked tasks such as
-Contains written and marked tasks,
tests, examinations
assignments, projects and
individually and in groups. These
interpretations were observed as two of the tests and examinations. There was
investigations, informal and formal -Learners’ performances in the
various tasks in statistical graph given
tests and examinations. Similar
assessment instruments with feedback
areas were learners had difficulties. There also evidence of intervention in
evidence of learner difficulties in
to them were indicated showing
from the learners contain similar
was also evidence of teaching intervention. learners’ learning difficulties in
statistical graphs was observed.
whether they perform well or not and
and formal tests and examinations done tests and examinations. Constructions and investigations, informal and formal
evidence of where the learners perform
constructing and interpreting graphs
Evidences of interventions to
what difficulties they may have
well and where their misconceptions
of grouped data such as scatter plots,
address the difficulties were also
encountered
and learning difficulties lie as shown in
histogram, ogive, etc.
observed.
the learner workbooks.
- learners had difficulties with the
construction and interpretation of
graph of grouped data.
Teacher
The teacher’s portfolios contain policy Teacher Bs’ portfolio contains similar
portfolios
documents such as the National
documents to that of teacher A.
Curriculum Statement (NCS), SBA and Intervention strategies similar to the ones
Teacher Cs’ portfolio contains similar There were similar documents and -Contains policy documents, tasks,
documents as those of teachers A and records in teacher Ds’ portfolio as tests, examinations with their
seen in teacher A, B and C’s
B.
memoranda in.
portfolios. Oral probing
other related assessment instruments.
used in case study 1 were used to address
Other documents are lesson plans
the misconceptions (e.g. drawing a
Similar intervention strategies used in questioning, pre-activities and
showing method used for teaching,
histogram instead of bar graph) and
case studies 1 and 2 were used to
how preconception were identified
learning difficulties about labelling the
address the misconceptions (drawing homework were used to identify
using oral questioning,, checking and
horizontal axis, observed by teacher B in
bar graph instead of histogram) and
marking of learners’ homework, work
case study 2 by analysing learners
learning difficulties in graph
schedules and records of assessment
classwork, homework, and assignment.
constructions of graph of grouped
Similar intervention strategies used
areas, how learners misconceptions
Oral probing questioning, pre-activities
data (eg give and scatter plots) as
in case studies 1, 2 and 3 were
(drawing bar graph instead of
and checking and marking of learners’
observed by teacher C in case study 3. used to address the misconception
histogram) and learning difficulties
homework were used to identify learners’ Oral probing questioning, pre-
((drawing bar graph instead of
(construction of graphs (labelling of
prior knowledge in histogram
histogram) and learning difficulties
activities and checking and marking
256
-Marks of learners in the tasks and
checking and marking of learners’ test already completed were available.
learners’ prior knowledge.
-Intervention strategies used to
identify learning difficulties such as
more activities days for afternoon
lessons were indicated. Learning
difficulties and misconceptions were
addressed by re-explanation of the
concept, extra tutoring and class
TEXTS
CASE STUDY 1
CASE STUDY 2
CASE STUDY 3
CASE STUDY 4
CODING
activities in statistics. The teachers
data axis)of grouped data such as
construction. There were also invitation
of learners’ homework were used to
of labelling data axis of grouped
histogram, ogive, scatter plots, etc.)
letters to mathematics workshops.
identify learners’ prior knowledge.
data observed by teacher D in case had invitation letters to several
study 4. Learning difficulties and
were identify by analysing learners
workshops in mathematics teaching.
misconceptions were addressed by
classwork, homework during marking,
Learning difficulties and
and the intervention strategies such as
misconceptions were addressed by re- re-explanation of the concept and
extra tutoring and class activities were
explanation of the concept and as
put in place to address them. There
Teacher A and C did. There were also extra tutoring and class activities
were also invitation letters to
invitation letters to mathematics
in statistics. There were also
mathematics workshops.
workshops.
invitation letters to mathematics
as Teacher A, B and C did using
workshops.
The SBA of teacher D contains
- contains contents of statistics
contents to be taught, assessment
topics to be taught assessment
curriculum, assessment tasks,
tasks, memoranda for the tasks and
tasks, memoranda for the tasks and memoranda for the tasks and formats
recording of learners’ performance in the
formats for grading and recording of
formats for grading and recording
for grading and recording of learners’
tasks.
learners’ performance in the tasks.
of learners’ performance in the
performance in the tasks.
School based
The SBA of teacher A contains topics
The SBA of teacher B contains content to The SBA of teacher C contains
assessment
to be taught, assessment tasks, and
taught assessment tasks, memoranda for
(SBA)
memoranda for the tasks and formats
the tasks and formats for grading and
for grading and recording learners’
performance in the tasks.
tasks.
Textbooks
Recommended
Recommended
Recommended
257
Recommended
APPENDIX XXII
An exercise in statistics for mathematics teachers
INSTRUCTION: This exercise is to get an insight into the basic knowledge that you
have about statistics teaching in school mathematics. Choose the option that best
represent the correct answer to each of the question and write it against the number of
the question. Do not write your name or the name of your school.
Duration: 40mins
Example
Find the mode of the following set of data: 4,3,5,4,5,6,4,6,5,4.?
A4
B3
C5
D6
E 4 and 3
ANSWER: A (The most occurring number in the set of data)
1
What would be the angle of sector representing the interval 10-20 in a pie chart in the
following frequency distribution table?
Interval
Frequency
A.
48 0
B.
72 0
C.
96 0
D.
120 0
E.
144 0
0-10
10-20
20-30
30-40
40-50
10
30
20
8
7
258
Use the frequency distribution table below to answer question
2
3
0-4
5-9
10-14
15-19
20-24
Frequency
3
5
7
4
1
Estimate the mode of the distribution.
A.
11.3
B.
11.5
C.
11.6
D.
12.0
E.
12.1
Calculate the mean of the distribution.
A
10.08
B
10.75
C
10.93
D
10.93
E.
4
Interval
11.79
The mean height of three groups of students consisting of 20, 16 and 14 students is
1.67m, 1.50m and 1.40m respectively. Find the mean height of all the students
A.
1,52m
B.
1.53m
C.
1.54m
D.
1.55m
E.
1.56m
259
5
The table below gives the frequency distribution of marks obtained by a group of
students in a test
Marks
3
4
5
6
7
8
Frequency
5
x-1
x
9
4
1
If the mean mark is 5, calculate the value of x.
6
A.
12
B.
13
C.
11
D.
9
E.
5
What is the median mark in question 4
A.
5
B.
4
C.
10
D.
5.5
E.
4.5
In a class, there are 80 students. The statistical distribution of the number of students offering
Physics, English, Mathematics, Hausa, and French is shown in a pie chart below.
Use this diagram to answer question 6.
260
7
8
How many students offer Mathematics?
A.
12
B.
13
C.
36
D.
10
E.
25
Calculate the percentage of student that offer English in the class.
A.
18.7%
B.
18.5%
C.
18.4%
D.
18.2%
E.
18.3%
The number of learners that were absent from a school in a class in a week were 2, 5, 6, 3 and
4. Use this information to answer question 9 and 10.
9
10
What is the variance of the data?
A.
3.0
B.
2.5
C.
5.0
D.
4.0
E.
2.0
Calculate the standard deviation of the distribution.
A.
2.4
B.
1.4
C.
3.4
D.
4.4
E.
5.0
The diagram below shows the number of HIV+ males and Females per age group in South
Africa in 2003. Use this information to answer questions 7 and 8.
261
11
12
How many South Africans were HIV+ in 2003?
A.
5542348
B.
5543248
C.
5554238
D.
5542384
E.
5524348
What percentage of male and HIV+ South Africans are in all the groups in 2003.
A.
44.50%
B. 4 4.19%
C.
44, 05%
D.
42.40%
E.
4.40%
262
13
If the population of South Africa is 46560400, how many South Africans are not
HIV+?
14
A.
41018052
B.
441018052
C.
41036052
D.
14018052
E.
5542348
Which age group is most infected with HIV/AIDS in 2003?
A.
(20-24) years
B.
40+ years
C.
(30-34) years
D.
(25-29) years
E. ( 15-19) years
The frequency distribution of marks of 800 students in an examination is display in a
cumulative frequency curve as shown below. Use the diagram to answer questions9 -11.
263
Marks (%)
15
Use your ogive to determine the 50th percentile.
A.
47.5
B.
43.5
C.
37.5
D.
57.5
E.
67.5
The candidates who score less than 25% are to be withdrawn from the institution,
while those that score more than 75% are to be awarded scholarship. Estimate:
16
The number of candidates that will be withdrawn from the institution.
A.
100
B.
80
C.
180
D.
200
E.
70
264
17
How many candidates will be retained in the institution but will not enjoy the
scholarship award?
18
A.
640
B.
560
C.
300
D.
440
E.
540
How many candidates will be retained in the institution but will not enjoy the
scholarship award?
A.
640
B.
560
C.
300
D.
440
E.
540
The graph below display the amount of time that a group of students spent in preparation for
a test and the marks that they scored in the test. The line of best fit has been included on the
scatter plot. Use this diagram to answer questions 12.
19
The graph shows a ------------------------------- correlation between the amount of time
the students spent in preparing the examinations and the marks scored.
265
20
A.
Strong
B.
No
C.
Weak
D.
Strong and weak
E.
None of the above
Determine the equation of the line of best fit for the scatter plot.
A`
Y = 0.22x + 40
B
Y = 0.50x + 40
C
Y = 0.70x + 40
D
Y = 22x + 40
E
Y = 2.2x + 40
Total marks: 100
266
APPENDIX XXIII
Memo for final conceptual knowledge exercise, march 2010
QUESTION
ANSWER
1
E
2
C
3
B
4
C
5
C
6
A
7
D
8
E
9
E
10
B
11
A
12
B
13
A
14
D
15
A
16
B
17
C
18
A
19
A
20
A
267
APPENDIX XXIV
Examining the content knowledge of mathematics teachers in statistics teaching
Duration: 30mins
The following are the topics to be taught in statistics under data handling in the new National
Curriculum Statements for grades 10-12: stem-and-leaf; mode, median and mean of ungroup
data; frequency table of group data; range, percentiles, quartiles; inter-quartiles and semiquartile range; bar and compound bar graphs; histogram; frequency polygons; pie charts; line
and broken line graphs; box and whisker plot; variance, mean deviation; standard deviation;
Ogives; five number summary; scatter plots; line of best fit.
a) Arrange the topics in each grade on how you think they should be taught in grades 10,
11 and 12.
b) With an arrow, show how you can teach these topics sequentially in each grade. For
example, you observe morning before afternoon and before evening. Therefore;
Morning
afternoon
evening
A)
GRADE 10
GRADE 11
GRADE 12
268
GRADE 10
GRADE 11
GRADE 12
269
APPENDIX XXV
Examining the content knowledge of mathematics teachers in statistics teaching
Duration: 30mins
The following are the topics to be taught in statistics under data handling in the new National
Curriculum Statements for grades 10-12: stem and leave; mode, median and mean of group
data; frequency table of group data; range, percentiles, quartiles; inter-quartiles and semiquartile range; bar and compound bar graphs; histogram; frequency polygons; pie charts; line
and broken line graphs; box and whisker plot; variance, mean deviation; standard deviation;
Ogives; five number summary; scatter plots; line of best fit.
c) Arrange the topics in each grade on how you think they should be taught in grades 10,
11 and 12.
d) With an arrow, show how you can teach these topics sequentially in each grade. For
example, you observe morning before afternoon and before evening. Therefore;
Morning
afternoon
evening
SOLUTION
A)
GRADE 10
GRADE 11
GRADE 12
Mode, Median, Mean, Ranges,
Five number summary, Box and
N/A
whisker diagrams, Ogives,
(ungrouped data), Frequency table,
Bar and Compound bar graphs,
Variance and Standard deviation,
Scatter diagrams, Lines of best fit
Histogram, Frequency polygons,
Pie charts, Line and broken line
graphs. Mode, median and mean
(grouped data), Quartiles, Interquartiles and semi-inter-quartile
range
270
GRADE 10
Mode
Median
Frequency table
Mean
Ranges ( Ungrouped data)
Bar and Compound bar graphs
Frequency Polygon
Histogram
Pie Charts Line and broken line graphs.
Mode
Median
Mean (Grouped data)
Quartiles
Inter-quartile and semi-inter-quartile ranges
GRADE 11
Five number summary
whisker diagrams
Box and whisker diagrams
Ogives
Scatter diagrams
1
A(1)
Box and
Variance and Standard deviation
Scatter diagrams
Grade 10 topics (25%)
Lines of best fit
Marks deducted
Marks
obtained
a
Missing one topic in the arrangement.
2
23
b
Missing two topics in the arrangement.
4
21
c
Missing three topics in the arrangement.
6
19
d
Missing four topics in the arrangement.
8
17
e
Missing five topics or more in the
10
15
arrangement.
A2
Grade 11 topics (25%)
Marks deducted
Marks obtained
a
Missing one topic in the arrangement.
2
23
b
Missing two topics in the arrangement.
4
21
c
Missing three topics in the arrangement.
6
19
d
Missing four topics in the arrangement.
8
17
e
Missing five or more topics in the
10
15
arrangement.
271
2
B1
Grade 10 with links (25%)
Marks Deducted
Marks obtain
a
Missing one topic in the arrangement.
2
23
b
Missing two topics in the arrangement.
4
21
c
Missing three topics in the arrangement.
6
19
d
Missing four topics in the arrangement.
8
17
e
Missing five or more topics in the
10
15
arrangement.
B2
Grade 11 with links (25%)
Marks Deducted
Marks obtain
a
Missing one topic in the arrangement.
2
23
b
Missing two topics in the arrangement.
4
21
c
Missing three topics in the arrangement.
6
19
d
Missing four topics in the arrangement.
8
17
e
Missing five or more topics in the
10
15
Grade 12 (N/A)
Marks Deducted
Marks obtain
N/A
Deduct 10 marks from
Balance after
total marks if any topic is
deduction
arrangement.
B3
written in grade 1 2. But
if the same topic in grade
11 is written in grade 12,
no mark should be
deducted. It should be
regarded as a revision in
grade 12.
272
APPENDIX XXVI
The interview schedule for mathematics teachers.
This interview probes the content knowledge in statistics and educational background
that may have enabled the teachers to develop their topic-specific PCK in statistics.
Time : 30 minutes
1)
Which university / college did you attend?
2)
What qualifications did you obtain?
3)
What course/subject/module did you study at the university/ college?
4)
How long did you study this course/subject?
273
5a)
If one of the courses in (3) is mathematics methodology, how did it help you to
prepare lessons for teaching?
5b)
How do you know that your teaching is effective?
6)
Do you have an interest in the teaching of mathematics? If yes/no, why?
7)
What is your understanding of the nature of the statistics you are teaching?
8)
Do learners understand the topic?
274
9)
Do learners enjoy the topic? If yes/no, why?
10)
In your own opinion and based on your experience in the teaching of statistics, how
do you see the topic (statistics) in mathematics?
11)
Do your learners understand your lessons based on the instructional approach for
teaching as recommended in the curriculum?
12)
If the learners have any problems in understanding the topic based on the
instructional approach, what do you do to help them to understand?
275
13)
What other instructional strategies do you use for teaching and why?
14)
What learning difficulties do you remember experiencing as a pupil and as a
university student or from teaching experience in statistics?
15)
Have you ever been to a mathematics workshop or teachers’ development
programme?
16)
If your answer in (15) is yes, what was the content of the workshop?
276
17)
What was the duration of the workshop?
18)
Were the workshop facilitators mathematics teachers or mathematics expert?
19)
As a mathematics teacher, what did you benefit from the workshop?
20)
Would you recommend that similar workshops be held for teachers in subsequent
time?
277
APPENDIX XXVII
Report on the teaching of statistics.
This schedule is to guide the mathematics teachers in written report during the four
weeks of teaching statistical graphs in grade 11. Any other relevant information may
be added by the teacher during the course of teaching.
Duration: 4 weeks
1) What learning difficulties do you identify in learners when teaching a topic?
2) What difficulties do you experience in the teaching of statistical graphs?
3) What do you find interesting in this topic and why?
4) What do you think you find less difficult to teach in the topic?
5) How did the learners respond to classroom activities as well as homework or
assignments?
278
6) What changes would you make next time with regard to the difficulties you
encountered while teaching, either on your part or on the part of the learners?
7) How do you identify the preconception and misconceptions of the learners during
teaching?
8) What preconceptions or misconceptions do you identify?
9) How would you address the preconceptions and misconceptions, if any, identified
during the teaching and learning process?
279
APPENDIX XXVIII
The questionnaire for mathematics teachers.
This questionnaire aimed at investigating what the teacher did while teaching statistical
graphs in grade 11.
Duration: 15 mins
1. How long was the lesson?
2. What was the topic of your lesson?
3. What were the objectives of your lesson?
4. What prior knowledge does your lesson require?
280
5. Did the learners have the prior knowledge (preconceptions) of the topic?
6.
How did you identify the prior knowledge (preconceptions) which the learners bring to
the class about statistical graphs?
7. Did you think the learners achieved the objective of the lesson?
8. How did the learners respond to class activities, homework and assignments?
9. Were you able to follow the lesson as planned to the end of the lesson?
281
10
What difficulties did the learners experience?
11
How did you address these difficulties?
12
How would you improve the lesson?
13
Do you normally evaluate your teaching?
14
What is it about statistics that makes the learning easy or difficult?
282
15
How do you evaluate your teaching performance?
16
For what reasons do you evaluate your teaching?
17
Were the students able to use the knowledge acquired to solve other problems?
283
APPENDIX XXIX
Instrument validation form for the conceptual knowledge exercise
Pedagogical content knowledge which is topic specific is conceptualised to include content
specific knowledge, content specific instructional strategies, conceptions and learners’
learning difficulties. The study participants (competent mathematics teachers) have
developed PCK and used it to assist learners to perform well in mathematics as evidence at
the senior certificate examination result for some period of years. This instrument is meant to
measure the content of a chosen topic (statistics) according to the National Curriculum
Statements (NCS) in mathematics, which the competent mathematics teachers have and
demonstrate in statistics teaching.
I therefore solicit few moment of your time to help me to validate the instrument using
SURENESS and RELEVANCE. By indicating sureness, one has no doubt that the
instrument measures the content of the chosen topic. By indicating relevance, one has no
doubt that the instrument is valuable and useful in measuring the content knowledge of the
chosen topic.
The rating levels for SURENESS are: 1 = not very sure; 2 = fairly sure; and 3 = very sure.
The rating levels for RELEVANCE are: 1 = low/not relevant; 2 = fairly relevant; and 3 =
highly relevant.
QUESTION
NUMBER
SURENESS
RELEVANCE
1
2
3
4
5
284
DO NOT WRITE ON
THIS COLUMN
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Comments:
________________________________
________________________________
Signature of Reviewer and qualification
Date
285
APPENDIX XXX
Ethical clearance certificate
286
APPENDIX XXXI
A sample of teachers’ response to concept mapping exercise
287
APPENDIX XXXII
LESSON OBSERVATION SHEET
DATE: .....................................................
DURATION OF THE LESSON: ................................................
The practical investigation lesson will be observed against the following attributes:
1)
2)
PLANNING
1.1
Lesson topic
1.2
Learning outcomes
1.3
Assessment Standards
1.4
Resources used
PEDAGOGICAL ISSUES
2.1
Introduction of the lesson
2.2
General class handling
2.2.1 Class organization
2.2.2 Discipline
2.2.3 Interactions
2.2.4 Movement
2.2.5 Learning climate
2.2.6 The involvement of the lesson
2,3
Lesson Development (Progression)
2.4
Consolidation of the lesson
2.5
Description of teaching and learning
2.5.1 Language
2.5.2 Questioning techniques
2.5.3
Assessments
2.5.4
The use of resources
2.5.5
Knowledge of the teacher
a) How did the teacher identify learners’ preconceptions, if any, in a
topic as indicated in the lesson plan? Did he or she demonstrate
knowledge of learners’ anticipated learning difficulties in the topic
during the lesson and in the lesson plan?
288
b) Did the teacher demonstrate his or her subject matter content
knowledge of the topic he or she was teaching?
c) What instructional skills and strategies did he or she use in teaching
the topic (statistics)?
2.5.6 Errors and misconceptions
d)
How did he or she identify the learners’ misconceptions and
learning difficulties in the topic he or she was teaching?
e) How did he or she address the identified misconceptions and learning
difficulties?
3)
LEARNER RELATED ACTIVITIES
4)
TEACHER RELATED ACTIVITIES
5)
EVALUATION / CONCLUSIONS
289
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