smp780 15378129 final ruan
TRANSCRIPT
GRADE 9 MATHEMATICS TEACHERS’ PCK REGARDING TEACHING AS REFLECTED IN THEIR PRACTICES
by
Ruan Kapp
Department of Science, Mathematics and Technology Education
Faculty of Education
University of Pretoria
Supervisor: Dr. JJ Botha
October 2015
© University of Pretoria
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SMP780 2015
Topic: Final Research Report
DECLARATION FORM FOR NON-PLAGIARISM
I (full names): Ruan Kapp
Student number: 15378129
Declaration:
1. I understand what plagiarism entails and I am aware of the University’s policy in this regard.
2. I declare that this assignment is my own original work. Where someone else’s work was used
(whether from printed source, the internet or any other source), due acknowledgement was given
and reference was made according to the departmental requirements.
3. I did not make use of another student’s previous work and submitted it as my own.
4. I did not allow and will not allow anyone to copy my work with the intention of presenting it as
his/her own.
Signature ____________________________________________________________
Date: 27 October 2015
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Table of Contents Title: .............................................................................................................................................................. 1
1. Introduction .............................................................................................................................................. 1
2. Problem statement ................................................................................................................................... 2
3. Rationale ................................................................................................................................................... 2
4. Research Question: ................................................................................................................................... 3
4.1. The main question that guides this study is: ..................................................................................... 3
4.2. Sub-questions: ................................................................................................................................... 3
5. Research Objectives .................................................................................................................................. 3
6. Literature review ....................................................................................................................................... 4
6.1. Introduction ....................................................................................................................................... 4
6.2. The school subject Mathematics in a South African context ............................................................. 4
6.3. Mathematics teacher’s knowledge .................................................................................................... 5
6.4. Overview of the different domains of teachers’ knowledge ............................................................. 5
6.4.1. Shulman’s (1986) categories of knowledge ................................................................................ 6
6.4.2. Hill, Ball and Schilling’s (2008) domain map for mathematical knowledge for teaching ........... 6
6.5. Teachers’ instructional practices ....................................................................................................... 8
6.5.1. Tasks ............................................................................................................................................ 9
6.5.1.1. Modes of representation ......................................................................................................... 9
6.5.1.2. Sequencing and difficulty levels ............................................................................................... 9
6.5.2. Learning environment ................................................................................................................... 10
6.5.2.1. Modes of instruction and pacing ........................................................................................... 10
6.6. Chapter summary............................................................................................................................. 11
7. Methodology ........................................................................................................................................... 11
7.1. Introduction ..................................................................................................................................... 11
7.2. Research paradigm .......................................................................................................................... 11
7.3. Research approach........................................................................................................................... 12
7.4. Research design ............................................................................................................................... 12
7.5. Research site and sampling .............................................................................................................. 13
7.6. Data collection techniques .............................................................................................................. 13
7.7. Data analysis strategies .................................................................................................................... 14
7.8. Quality assurance criteria ................................................................................................................ 14
7.9. Trustworthiness of the study ........................................................................................................... 14
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7.10. Validity and reliability of the study ................................................................................................ 15
7.11. Ethical considerations .................................................................................................................... 15
7.12. Summary ........................................................................................................................................ 16
8. Presentation of the results ..................................................................................................................... 16
8.1. Introduction ..................................................................................................................................... 16
8.2. Data collection process .................................................................................................................... 16
8.3. Data analysis strategies .................................................................................................................... 17
8.3.1. Transcribing the data ................................................................................................................ 17
8.3.2. Coding the data ......................................................................................................................... 17
8.4. Information regarding the two participants .................................................................................... 22
8.4.1. The school ................................................................................................................................. 22
8.4.2. Elize ........................................................................................................................................... 23
8.4.3. Alisha ......................................................................................................................................... 23
8.5. Lesson dimension 1: Tasks ............................................................................................................... 24
8.5.1. Elize’s instructional practices: Tasks ......................................................................................... 24
8.5.2. Alisha’s instructional practices: Tasks ....................................................................................... 28
8.6. Lesson dimension 2: Learning environment .................................................................................... 31
8.6.1. Elize’s instructional practices: Learning environment .............................................................. 31
8.6.2. Alisha’s instructional practices: Learning environment ............................................................ 31
8.7. Findings from interviews .................................................................................................................. 33
8.7.1. Elize’s interview ........................................................................................................................ 33
8.7.2. Alisha’s interview ...................................................................................................................... 34
8.8. Conclusion ........................................................................................................................................ 34
9. Conclusions and Implications .................................................................................................................. 35
9.1. Introduction ..................................................................................................................................... 35
9.2. Summary of the sections ................................................................................................................. 35
9.3. The research questions .................................................................................................................... 36
9.3.1. To what extent did the teachers base new knowledge on the learners’ prior knowledge? .... 37
9.3.2. Which forms of representation did the teachers use? ............................................................. 37
9.3.3. How can the sequencing of content by the teachers during the lesson be described? ........... 38
9.3.4. How appropriate were the teaching strategies used by the teachers? .................................... 39
9.4. Summary of my findings .................................................................................................................. 40
9.5. What would I have done differently? .............................................................................................. 40
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9.6. Limitations of the study ................................................................................................................... 40
9.7. Conclusion ........................................................................................................................................ 41
9.8. Possible implications of the findings ................................................................................................ 41
9.9. Recommendations for future study ................................................................................................. 42
References .................................................................................................................................................. 43
Appendices .................................................................................................................................................. 48
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List of acronyms
CAPS
CCK
DBE
FET
GET
KCS
KCT
MCK
MKT
ML
PCK
TIMSS
UP
Curriculum and Assessment Policy Statement
Common Content Knowledge
Department of Basic Education (South Africa)
Further Education and Training
General Education and Training
Knowledge of Content and Students
Knowledge of Content and Teaching
Mathematical content knowledge
Mathematical Knowledge for Teaching
Mathematical Literacy (the subject)
Pedagogical content knowledge
Trends in International Mathematics and Science Study
University of Pretoria
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List of figures
Figure 6.1 Hill, Ball and Schilling’s (2008) domain map for mathematical
Knowledge for teaching
13
Figure 8.1 Mathematics teachers’ instructional practices: Tasks 26
Figure 8.2 Mathematics teachers’ instructional practices: Learning
environment
27
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List of appendices
Appendix A Letter of consent to the Mathematics learners
Appendix B Letter of consent to the principal
Appendix C Letter of consent to teachers
Appendix D Ethical clearance certificate
Appendix E Observation sheet for observing Mathematics teachers’ lessons
Appendix F Interview schedule
Title:
Grade 9 Mathematics teachers’ PCK regarding teaching as
reflected in their practices
1. Introduction
In many countries around the world there is a general belief that to live a “normal life”
requires the use of mathematics on an everyday basis (Dörfler & Mclone, 1986, p. 49).
This viewpoint is supported by the South African school curriculum as it states that:
“Mathematical problem solving enables us to understand the world (physical, social and
economic) around us, and, most of all, to teach us to think creatively” (Department of
Basic Education, 2011). Dörfler and Mclone (1986) continue by saying that today there is
a heavy emphasis on the importance of mathematics and a worldwide belief that children
should excel in their knowledge and skills in this subject. From my experience as a
teacher I have also experienced that the subject mathematics is heavily emphasised at
school level. These expectations place a big responsibility on mathematics teachers and
it begs the question as to what skills and practises create a good mathematics teacher?
Adedoyin (2011) suggested that good mathematics teachers should possess apart from
basic content knowledge, a substantial amount of specialised knowledge which is known
as pedagogical content knowledge (PCK). At the PCK summit in Colorado, Springs, USA
in October 2012, a group of experts on PCK proposed a description that entitles PCK to
be a personal attribute of a teacher, considered both a knowledge and an action (Gess-
Newsome & Carlson, 2013). They believe that it is: “the knowledge of, reasoning behind,
planning for, and presentation of teaching a specific topic in a specific way for a specific
reason to specific students for enhanced student outcomes” (Gess-Newsome & Carlson,
2013). Since PCK consists of multiple domains of knowledge, it is not possible to address
all these aspects in a study of limited scope. This study will look at a teacher’s knowledge
of how to teach within a classroom setup. It seeks to examine senior phase mathematics
teachers’ PCK regarding teaching as it is reflected in their classroom practises.
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2. Problem statement
Nationally senior phase learners are underachieving or not performing well in
mathematics. This is evident from the poor Annual National Assessment (ANA) results
achieved over the last few years. In 2014 the ANA Grade 9 average for mathematics was
a shocking 10.9% (Department of Basic Education, 2014). The Trends in International
Mathematics and Science Study (TIMSS) is an international assessment of Grade 8
learners’ mathematics and science content knowledge. TIMSS is administered every 4th
year and in 2003 South Africa was listed last. For 2007 there is no results as we did not
partake in the study and in 2011 we improved to take up second last place bearing in
mind that the tests was administered to Grade 9 learners (Reddy, et al., 2012). These
results support the above claim. The poor performance could be due to a number of
reasons; according to Spaull (2013) one specific reason may be associated with the
content knowledge and more specifically the in-depth PCK of teachers. According to
Shulman (1986) a teacher needs proper subject matter knowledge and a high level of
pedagogical content knowledge to assure effective teaching.
3. Rationale
A rationale addresses how the researcher developed an interest in the topic and why the
study is worth doing (Vithal & Jansen, 1997). As a trained mathematics teacher, but
currently in teacher education, I can relate to the factors that the young mathematics
teachers entering into the teaching profession must deal with. Working alongside my
senior teachers, I see the differences in teaching beliefs, knowledge, approaches and
methods. I therefore developed a personal interest in the extent to which teachers’ PCK
contributes to their classroom practises. Engaging with teachers at subject meetings I
noticed how the more experienced ones came across as confident and in control of the
situation; whilst I faced the many new challenges of a young teacher. These challenges
for me included aspects such as pacing an overfull curriculum, classroom discipline,
subject knowledge and preparing a good lesson. Understanding how teachers’ PCK
influences their teaching is thus a personal interest of mine as Koellner, Jacobs, Borko,
Schneider, Pittman and Eiteljorg, (2007) mention that no factor is more important than the
teacher in achieving the vision for school mathematics.
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4. Research Question:
4.1. The main question that guides this study is:
How can Grade 9 Mathematics teachers’ PCK regarding teaching as reflected in their
practices be described?
To answer the main question, the following sub-questions are stated:
4.2. Sub-questions:
1. To what extent did the teachers base new knowledge on the learners’ prior
knowledge?
2. Which forms of representation did the teachers use?
3. How can the sequencing of content by the teachers during the lesson be
described?
4. How appropriate were the teaching strategies used by the teachers?
5. Research Objectives
The objective of this research is to gain a deeper understanding into Grade 9 mathematics
teachers’ PCK regarding teaching. If we could describe how teachers’ PCK influences the
lesson planning process we could gain significant grounds in pre-service teacher training.
The research attempts to describe teachers’ PCK by looking at several factors such as
content sequencing, activating a learner’s prior knowledge, use of appropriate teaching
strategies and various representations. Once we are able to formulate a successful
description it could be used to improve many future teachers’ abilities and approaches to
quality education.
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6. Literature review
6.1. Introduction
This section focuses on the literature review, which attempts to address the theoretical
issues related to pedagogical content knowledge (PCK) as one of the knowledge domains
of a mathematics teacher. The literature review gives a background of mathematics as a
school subject in the 21st Century worldwide, as well as in South Africa. An overview is
given of different domains of teachers’ knowledge with specific reference to Hill, Ball, and
Schilling (2008) as it forms part of the conceptual framework. Teachers’ instructional
practises are also discussed in depth with an emphasis on the framework of Artzt,
Armour-Thomas, and Curcio (2008).
6.2. The school subject Mathematics in a South African context
In the 21st Century, there is an ongoing drive to make the subject mathematics accessible
to all people. According to Adler, Ball, Krainer, Lin and Novotna (2005), mathematics is
globally viewed as not just a necessary subject, but a necessary skill that is required for
social responsibility, and what we are witnissing now is a “massification” of this school
subject. In South Africa there has also been a determination to make mathematics
accessable to all learners. An article from the Deparment of Basic Education’s (2015)
website states that: “Whereas mathematics was not compulsory under the previous
system, under the new system all candidates must take either Mathematics or
Mathematical Literacy” In the Further Education and Training band (Department of Basic
Education, 2015). Therefore making any form of mathematics accessible to all learners
has consequences.
Adler, et al. (2005) argue that an increase in the demand for mathematics proficiency also
has an increase in the need for quality mathematics teaching. More teachers and quality
mathematics teaching is thus a prerequisite if we wish to increase mathematics
proficiency amongst the majority of South Africans. There are many aspects that influence
quality mathematics teaching in South Africa. One such aspect is the number of learners
in classes. According to Setati and Adler, (2000, p. 243), South African mathematics
classrooms are relatively large with more than 35 learners in a single mathematics class
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and in rural areas this number is likely to increase. Quality Mathematics teaching requires
that learners receive as much individual attention as possible and large classes
complicate the teacher’s role of attending to all learners’ needs. Another aspect
influencing the quality of mathematics teaching in South Africa, is the language of learning
and teaching (LOLT) as many South African learners receive schooling in their second or
even third language (Setati & Adler, 2000, p. 243). Research shows that mother tongue
education is crucial for mathematical development (Adler, 1995; Austin & Howson, 1979;
Mji & Bothes, 2010)
6.3. Mathematics teacher’s knowledge
The most fundamental aspect however in effective and proficient mathematics teaching
is a high level of knowledge (Kilpatrick, 2001; Taylor, 2008). A common finding across
studies shows large numbers of South African mathematics teachers who lack
fundamental and conceptual understanding of mathematical concepts (Carnoy,
Chisholm, & Chilisa, 2012; Taylor & Taylor, 2013; Venkat & Spaull, 2015). Teachers
cannot help learners with content clarification if they do not understand the content
themselves. A teacher needs proper subject matter knowledge and a high level of
pedogogical content knowledge (PCK) to assure effective teaching (Shulman, 1986; Ma,
1999).
According to Venkat and Spaull (2015) there is an agreement amongst studies that PCK
rests firmly on a well-developed content knowledge base. PCK is considered as
knowledge that is unique to teachers; knowledge that can only be developed over time
through experience in the classroom or practice and can therefore not be taught (Ball,
1988; Ball et al., 2005; Koellner et al., 2007; Ma, 1999; Shulman, 1986). This study is
interested in understanding of the PCK that Grade 9 mathematics teachers illustrate
through their instructional practises.
6.4. Overview of the different domains of teachers’ knowledge
Taylor (2008) found that teachers in South Africa clearly do not have the knowledge that
the curricula require to proficiently teach the learners. To address this problem of
teachers’ inadequacy, the school system has to re-establish the emphasis on expert
knowledge. This section discusses some of the research on teachers’ domains of
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knowledge. Shulman’s (1986) categories of knowledge as well as Hill, Ball and Schilling’s
(2008) domain map for mathematical knowledge for teaching are now described.
6.4.1. Shulman’s (1986) categories of knowledge
Shulman (1986) indicated three categories of knowledge a mathematics teacher needs.
These categories are: “subject matter content knowledge, pedagogical content
knowledge (PCK) and curricular knowledge”. Botha, (2011) explains that subject matter
content knowledge goes beyond knowledge of the facts or concepts of a domain to
understand the structures of the subject matter. According to Shulman (1987), the second
category of knowledge namely PCK, depends on how the teacher transforms subject
matter knowledge into various forms that enable students in different learning
environments to understand the subject matter. PCK includes teachers’ knowledge of
various analogies, representations, explanations, demonstrations and will later be
discussed in more detail. The third category of knowledge is curricular knowledge, which
refers to the knowledge of curricular materials that teachers use to teach specific topics
and ideas to a particular group of learners (Cogill, 2008). It requires understanding of
children’s learning potential, as well as knowledge of national syllabi for previous and later
years taught and school planning documents.
6.4.2. Hill, Ball and Schilling’s (2008) domain map for mathematical knowledge for
teaching
Researchers (Ball, Thames & Phelps, 2008; Hill, et al., 2008; Delaney, Ball, Hill, Schilling
& Zopf, 2008) extend Shulman’s construct of PCK. These researchers use a model
(Figure 1) which links their domains of content knowledge for teaching onto two of
Shulman’s (1986) initial categories for PCK, those of subject matter knowledge and
pedagogical content knowledge.
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Hill et al. (2008) as indicated in Figure 1 add additional subdomains under each of
Shulman’s (1986) domains. Under subject matter knowledge they include: 1) Common
Content Knowledge (CCK); 2) Specialised Content Knowledge (SCK); and 3) Knowledge
at the mathematical horizon. According to Hurrell, (2013, p.57), Common Content
Knowledge is mathematical knowledge and skills that is used not only for the teaching
discourse, but in any general setting. Specialised Content Knowledge explained by Botha,
(2011) is the mathematical knowledge used in the teaching profession. Teachers need to
be able to explain why certain mathematical concepts work the way they do. Knowledge
at the mathematical horizon is described by Ball, et al. (2008) as that knowledge where a
teacher has to identify how the topics of mathematics they teach at this point fit into the
mathematics which comes at a later stage.
Under pedagogical content knowledge Hill et al. (2008) include: 1) Knowledge of content
and student (KCS); 2) Knowledge of content and curriculum (KCC) and 3) Knowledge of
content and teaching (KCT). According to Hurrell, (2013, p. 57), Knowledge of content
and students (KCS) is the knowledge about the level of the learners and the mathematics
to be taught. Knowledge of content and curriculum is explained as the knowledge of the
teacher regarding the structure of the mathematics curriculum (Hurrell, 2013). The third
Figure 6.1 Mathematical knowledge for teaching (Hill et al., 2008, p. 377)
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domain of knowledge namely Knowledge of the content and teaching (KCS) is a
amalgamation of knowing about teaching and mathematics (Hurrell, 2013, p. 57). This
domain includes aspects such as the sequencing of mathematical content and the
selection of appriopriate representations to explain and illustrate the content. This study
is based on the PCK domain that focuses on the knowledge of content and teaching as it
tries to examine Grade 9 mathematics teachers’ PCK regarding teaching as it is reflected
in their classroom practices.
6.5. Teachers’ instructional practices
There are many pedagogical aspects associated with the process of teaching and
learning. This study is however interested in the instructional practices of the teacher
within the classroom. According to Artzt et al. (2008) it is inside the classroom where
teachers’ goals, knowledge and beliefs are the driving forces behind their instructional
efforts to guide and tutor learners in their acquisition of knowledge. The term ‘instructional
practice’ best describes the focus of this study being Grade 9 mathematics teachers’
actions in presenting their lessons. There are different views concerning the components
of a teacher’s instructional practice. Artzt et al. (2008) advocates the use of a phase
dimension framework to examine a teacher’s instructional practice. The framework is
guided by three observable aspects of a mathematical lesson, namely tasks, discourse,
and the learning environment.
Franke, Kazemi, and Battey (2007) in their framework make use of discourse, norms and
building relationships as the three features of classroom practices. Botha (2011, p. 54)
concludes that from these two views the teachers’ practices could be illustrated as a social
environment. Every stakeholder in the classroom is in a relationship with one another,
where they have the opportinity to widen their knowledge through communicating and
engaging in challenging tasks. This follows the research paradigm of social constructivism
which suggests that all knowledge is constructed and not only based upon prior
knowledge, but also includes the cultural and social context (Ollerton, 2009). For the
purpose of my study the two observable dimensions of tasks and learning environment
as discussed by Artzt et al. (2008) are used, since they address the practical issues of
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classroom practice in this scenario. What follows is a brief discussion of the important sub
dimensions under the two dimensions of tasks and learning environment.
6.5.1. Tasks
According to Artzt et al. (2008) the purpose of tasks within the classroom is to provide a
platform for learners to engage in significant problem solving and connect their prior
knowledge to new information. Tasks handed to learners must be formulated in such a
way that it connects the classroom mathematics with that of real world problem-solving.
The researcher is particularly interested in two sub dimensions and they are: 1) Modes of
representation and 2) Sequencing and difficulty.
6.5.1.1. Modes of representation
Modes of representation according to Artzt et al. (2008, p.12) are the practices for
representing mathematical concepts through the use of oral or written language,
diagrams, graphs, manipulatives, computers, or calculators. The National Council of
teachers of Mathematics (NCTM) has advocated the importance of technology in the
mathematics classroom and state that: “technology is essential in the teaching and
learning of mathematics; it influences the mathematics that is taught and enhances
student learning” (Kaput, 1992). According to Bransford, Brown and Cocking (2000), there
are also those who believe that technology is a waste of money and time whereas others
regard the presence of computer technology in schools as enhancing the learning in the
school. The recent interest in open source mathematical software and cellular phone
applications has made it easier and affordable for schools and teachers to acquire.
Mathematical software like GeoGebra could enhance the teaching of mathematics topics
such as geometry and functions.
6.5.1.2. Sequencing and difficulty levels
Artzt et al. (2008, p. 13) explain that the sequencing and difficulty levels of tasks must
allow students to make use of their prior knowledge and experiences to help them
understand the requirements of the tasks at hand. According to Kilpatrick et al. (2001) the
quality of teaching depends on whether the teachers select cognitively demanding tasks,
and whether these tasks unfold in such a way that it allows the student to elaborate and
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learn through the task. This point is emphasized by Bransford et al. (2000) who mention
that tasks must be set at an appropriate level of difficulty so that learners would stay
motivated. If the task is too easy, learners might become bored whilst a too difficult task
causes frustration amongst the learners.
6.5.2. Learning environment
The foundation for a learning environment in this study is based on the work of Artzt et
al. (2008). They state that a learning environment comprises a particular social and
intellectual climate, the use of effective modes of instruction and pacing of the content
and attending to certain administrative routines. My study is however only interested in
the modes of instruction and pacing that the teacher uses as it is directly linked to a
teacher classroom practices.
6.5.2.1. Modes of instruction and pacing
Included here are teaching strategies that teachers use in the classroom to help learners
attain the objectives of the lesson (Artzt et al., 2008). In general, different kinds of
instructional strategies, representations and activities are used in teaching mathematics.
Knowledge of instructional strategies requires understanding ways of representing
specific concepts, in order to facilitate student learning. Representations include
illustrations, examples, models, and analogies. Each instructional strategies has a
conceptual advantage and disadvantage over others (Ibeawuchi, 2010). According to
Wood and Sellers (1997) the teaching of mathematics concepts and skills should be
structured around problems to be solved. Learners should also be encouraged to work
co-operatively with each other in the classroom (Johnson & Johnson, 1975). Other
researchers propose the use of discussions and group work within the mathematics
classroom (Venkat & Graven, 2008).
All learning activities should be paced in such a way that learners have sufficient time to
participate and construct new knowledge. Hence PCK in this area includes attentiveness
of the relative strengths and weaknesses of a particular instructional strategy. PCK of this
type incorporates teachers’ knowledge of the conceptual power of a particular activity
(Magnusson et al., 1999). For any strategy to be powerful, the teacher must know the
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learners’ prior knowledge about a particular topic, and the possible difficulties they will
experience during the teaching and learning of the topic.
6.6. Chapter summary
In this section, the notion of different domains of knowledge and in particular PCK were
discussed with reference to the work of many scholars. With this study’s focus on
teachers’ PCK regarding content and teaching (Hill et al., 2008), different pedagogical
approaches, strategies and representations, use of meaningful sequencing of content and
appropriate instructional material were discussed. Furthermore, the framework for this
study is based on the domain map for mathematical knowledge for teaching and the
categories of an instructional practice, namely tasks and learning environment (Artzt et
al., 2008).
7. Methodology
7.1. Introduction
In this section the research methodology that was used to investigate the research
problem is described with reference to the research paradigm, approach and design. The
research site, sample selection and data collection techniques are carefully described as
well as the data analysis strategies. Lastly I discuss critical issues such as the
trustworthiness of the study and ethical considerations applicable to the study.
7.2. Research paradigm
A social constructivism paradigm was chosen for this study. This paradigm suggests that
all knowledge is constructed and based upon not only a learners’ prior knowledge, but
also the cultural and social context (Botha, 2011). A social constructivist philosophy of
mathematics is based on the fact that knowledge is not passively received, but actively
built (Ollerton, 2009). According to Ollerton (2009), most people do not operate
individually for the bulk of time, but in the classroom setup students might be encouraged
to work individually at first and then later share and compare their information. I strongly
share this view as I believe learners need to construct their knowledge through
meaningful problem-solving by engaging with each other as this is what happens most of
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the time in the cooperate world. Social interaction, group work, problem solving and
learner-centred approaches play significant roles in learners’ construction of their own
knowledge. The teacher must therefore guide and mentor the learners in developing their
own understanding.
7.3. Research approach
The research approach for this study is qualitative. A qualitative research approach is
appropriate as the researcher endeavours to describe an event in the social world from
the viewpoint of the individuals who are part of the ongoing event (Sinkovic, Penz, &
Ghauri, 2008). According to Hogan, Dolan, and Donnelly, (2009), qualitative research is
about researching specific meanings, emotions and practices that arise from the
interactions between participants. Therefore, in this study, the research interest is to study
the PCK of the teacher regarding their way of teaching as it is held by the participating
teachers. This will be done by observing crucial interactions with learners and lesson
presentations as the teachers teach various mathematics lessons. The qualitative
research approach in education, according to Mason (2010), allows the researcher to
understand and explore the richness, depth, context and complexity within which teachers
in the research context operate. As the interactions between teacher and learner vary
every time, qualitative research gives the researcher the ability to understand each
interaction independently.
7.4. Research design
This is a case study and according to Edwards and Talbot (1999, p. 51), the idea of a
case study is to allow a fine-tuned exploration of complex sets of inter-relationships. In
order to gain insight into this phenomenon of how teachers’ PCK influences their
classroom practises, an in-depth study of a small number of teachers are required and
that is why the case study method is chosen. The case study method allows the
researcher to gain insight into the PCK of the teachers as they teach their lessons. The
Grade 9 mathematics teachers are regarded as the ‘unit’ that is studied in order to
determine their PCK through their instructional practices. The researcher’s involvement
in the case study gives a sense of being there (Cohen, Manion, & Morrison, 2001, p. 79).
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7.5. Research site and sampling
A case study involves collecting high quality data. Edwards and Talbot (1999) point out
that the inductive approach requires small sampling that is information-orientated and at
the same time also representative. Due to the small scale nature of this research, the
sample consists of two South African Grade 9 Mathematics teachers at an urban school
in the Tshwane metropolitan. For this reason it is also not possible to choose a
representative sample. The sampling process is both convenient and purposeful. The
sampling is partly convenient as the school was chosen from schools in Tshwane that
were easily accessible. Two Grade 9 mathematics teachers will be purposefully chosen.
The participating teacher must have at least 1 year teaching experience. Grade 9 is
chosen because of the logical assumption that a larger variation in learners’ mathematical
ability would be plausible. The Senior Phase in the GET band concludes with Grade 9.
As previously mentioned, Mathematics is compulsory for all learners in the GET band in
South Africa.
7.6. Data collection techniques
Case studies tend to be time-consuming because the focus is on meanings and the
complexity of all the interrelations that exists (Edwards & Talbot, 1999). The advantage
however of case study research is the fact that multiple sources of data collection can be
used (Verwey, 2010). Consequently, two sources of data collection namely observations
and interviews, will be used in this research project. The method of data collection
consisted of two observations in two consecutive days. These observations of the teacher
will be done in an effort to obtain a relatively true account of the teacher’s instructional
practices. The type of observation I will use is that of the observer as participant
(Nieuwenhuis, 2007). The purpose of the classroom observations is to be able to describe
the teachers’ instructional practices according to the tasks handed to learners as well as
the explanations given for questions that learners might have. For this purpose an
observation schedule will be compiled based on the conceptual framework and research
questions. The observations will be audio recorded so that minimal disruption for the
learners can be assured and an observation sheet based on the framework will be used
on which field notes can be made.
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A written, semi-structured interview, in the format of a questionnaire, with the questions
formulated and organized in advance, will be conducted the period before the lesson.
According to Nieuwenhuis (2007, p. 87), the aim of qualitative interviews is to see the
world through the eyes of the participants and to learn more about the participants’
behaviours, beliefs and knowledge views. The two interviews will be based on the
teacher’s planning of the lessons, with specific reference to the way they teach (such as
the strategies they have chosen etc.) in order to gain insight in their PCK regarding
teaching.
7.7. Data analysis strategies
The observations for this study will be digitally voice recorded. The digital voice-
recordings shall be downloaded onto a computer. These recordings will be used in
conjunction with the researcher’s field notes to evaluate the teachers’ responses to
learners’ questions. It gives the researcher the ability to describe the nature of the
teacher’s PCK as information from interviews can be compared with new findings from
observations. A limitation of the digital voice-recordings however might be the inaudibility
of some of the learner contributions. All the contributions will also be transcribed.
Interviews will also be digitally voice recorded. Participants will be asked questions and
answers will be categorised accordingly. I will use DEDUCTIVE-inductive (uppercase
denotes the preference given to the style of analysis) qualitative data analysis as my
analysis will initially be deductive and then inductive. My raw data will be analysed
according to the categories that have been identified in the conceptual framework.
7.8. Quality assurance criteria
In this study I consider the different quality assurance aspects that are linked to qualitative
research. What follows is a description of aspects such as the trustworthiness and the
validity and reliability.
7.9. Trustworthiness of the study
According to Nieuwenhuis (2007, p.80), qualitative researchers use the term
trustworthiness when they are referring to research that is credible and trustworthy.
Although this research study makes use of a small sample size and the number of lessons
15
observed are also small; the trustworthiness of this study is still enhanced through the
use of several data collection strategies such as multiple observations and semi-
structured interviews.
7.10. Validity and reliability of the study
Internal validity according to Lincoln and Guba, (1985) is a degree of consistency between
the research phenomenon and the research findings: “What is being observed are
people’s constructions of reality, how they understand the world.” (Merriam, 1991, p. 167).
On the other hand, reliability refers to the consistency and re-applicability over time, over
instruments and over groups of respondents (Cohen et al., 2001, p. 117). I attended the
learning periods as a direct observer, while giving undivided attention to the classroom
events. My mathematical background and past experience of the classroom setup,
enhanced the persistent quality of the observations (Lincoln & Guba, 1985). I enhanced
the reliability of my study by consequently observing the teachers every time I visited the
classrooms (Cohen et al., 2001, p. 119).
7.11. Ethical considerations
Ethical clearance was obtained from the Ethics Committee in the Faculty of Education at
the University of Pretoria (UP) as well as from the DBE, prior to the commencement of
my research. The purpose for granting permission to the researcher for creating the roles
of observer and interviewer is to collect data (McMillan & Schumacher, 2001). All parties
involved in my research will be approached, including the headmaster, head of
mathematics at the school and the participating teachers. The participants will be invited
to take part in the study and they will be informed of the purpose of the study. Aspects
that will be addressed include: voluntary participation, informed consent, confidentiality,
anonymity and risk. There was no obligation to take part in the study an after joining the
study, the participants signed a letter of informed consent. The letter explained the
purpose of the study, the procedures to be followed as well as the advantages and
disadvantages.
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7.12. Summary
In this section I discussed the social constructivism philosophy as my research paradigm.
I also motivated my choice of qualitative research and the use of an exploratory case
study. Mention was also made of the research site and sampling. The data collection
instruments and process as well as the analysing strategies were discussed coupled with
quality assurance and ethical considerations.
8. Presentation of the results
8.1. Introduction
In this chapter I will briefly report on the data collection process as well as the data
analysis strategies. This is followed by an explanation regarding the coding of the data
based on my study’s conceptual framework. I also present the findings from each
participant involved in this study.
8.2. Data collection process
The data collection took place in Pretoria during the third quarter (August) of 2015. I setup
a meeting with the principal of the school as well as the head of department for
mathematics to discuss my study and request their participation. At the meeting a letter
of consent was handed to the principal and the two Grade 9 teachers that were identified
by the head of department to participate. The two participants1 were Elize and Alisha,
both from the same school. During the data collection period all communication and
arrangements were made through the head of department.
I kept to the data collection process2 of two observations with one interview conducted at
the end of the second observation. The duration of the interview was approximately 35
minutes. I only observed Grade 9 mathematics lessons and all participants had at least
one year experience of teaching Grade 9 mathematics3.
1 Pseudonyms were used for ethical purposes 2 The data collection process is discussed in Section 7.6: Data collection techniques. 3 The other selection criteria are discussed in Section 7.5: Research site and sampling
17
In Table 8.1 a timeline is given indicating the dates on which both participant’s lessons
were observed and interviews conducted.
Table 8.1: Timeline of the data collection process
Data gathering instrument Participants Date
Observation 1 Elize & Alisha 19 August 2015
Observation 2 Elize & Alisha 26 August 2015
Interview 1 Elize & Alisha 26 August 2015
8.3. Data analysis strategies
In Section 7 of the study, the DEDUCTIVE-inductive approach and analytic strategies
used in analysing the data were discussed. In this section however I only discuss the
transcribing and coding of the data. The inclusion and exclusion criteria for coding the
data are also given in table form 8.3.
8.3.1. Transcribing the data
I transcribed my video data verbatim to text directly after the data had been collected. For
both teachers I also had to translate the data from Afrikaans to English. After each
observation all hand-written field notes made during the observations as well as insights
that were thought of afterwards which had not been noted were written on a template
form. Uncertainties that emerged were cleared by watching the videos again.
8.3.2. Coding the data
I used a deductive approach based on my conceptual framework for coding the data. The
main theme of mathematics teachers’ instructional practices together with the subthemes
were chosen according to the work of Artzt et al. (2008). According to the raw data
analysed, I ascribed codes to the different lesson dimension indicators of each subtheme
by using the software program Atlas.ti 6.2, I coded the transcripts according to a set of
pre-determined categories of the lesson dimension indicators. These codes are given in
Table 8.2 that follows below. After the data were coded I created coding families which
are according to Archer (2009) clusters comprising codes that are related to one other.
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According to my conceptual framework, families were created by selecting from the list of
all codes those codes that were related to the subthemes.
8.3.2.1. Main theme: Mathematics teachers’ instructional practices
For this study there are two subthemes identified which could best describe the teachers’
instructional practices namely tasks and learning environment (Artzt et al., 2008). The
first column in Table 8.2 below indicates the two subthemes or lesson dimensions with
their different categories. In the second column are the descriptions of the lesson
dimension indicators with the codes created for them. All data were collected from the
observations only.
19
Table 8.2: Lesson dimensions and dimension indicators as inclusion criteria for coding the data (Adapted from Artzt et al., 2008)
LESSON DIMENSIONS DESCRRIPTION OF CODES
Tasks
Building on pre-knowledge TPK1. There should be a logical flow in the lesson such as revising prior knowledge before introducing new content.
Modes of representation
TMR1. Uses representations such as oral or written language, symbols, diagrams, graphs, tables, manipulatives, and computer or calculator representations to accurately facilitate content clarity.
TMR2. Provides multiple representations that enable learners to connect their prior knowledge and skills to the new mathematical situation such as graphs, tables, formulae.
Sequencing of content
TSC1. Sequences tasks and learning activities so that learners can progress in their cumulative understanding of a particular content area and can make connections between ideas learned in the past and those they will learn in the future such as working from easy to difficult and known to unknown tasks.
TSC2. Uses tasks, including homework that is suitable to what the learners already know and can do and what they need to learn or improve on. Tasks should involve past work, reinforce current work and set the stage for future work such as tasks where opportunity is given to practice identified or predicted learners’ misunderstandings.
Learning environments
Modes of instruction/pacing
LEIP1. Uses various instructional strategies that encourage and support student involvement as well as facilitate goal attainment such as cooperative learning, learners explaining work at the board, direct instruction (lecturing), abstract procedural, group work, active learning, discussion, problem solving, inquiry and team-teaching.
Source: Adapted from: A Cognitive Model for Examining Teachers' Instructional Practice in Mathematics: A Guide for Facilitating Teacher Reflection, by A.F. Artzt and E. Armour-Thomas, 1999, Educational Studies in Mathematics, 40(3), p. 217. Copyright © 1999 by Kluwer Academic Publishers. Adapted with kind permission from Kluwer Academic Publishers.
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Using Atlas.ti 6.2, networks of the code families, according to Archer (2009) are now
illustrated and explained. The data for the coding families were collected from the
lesson observations. The code family created for the first subtheme Tasks appear in
Figure 8.1 below. The broken line arrows indicate the sub categories under the lesson
dimension linked to the code family Tasks. Atlas.ti 6.2 uses solid line arrows with
double equal signs to indicate the codes associated with the different lesson
dimensions (Archer, 2009). For example codes TMR1 and TMR2 are associated with
lesson dimension Tasks: Modes of representation. A full description of each code such
as TMR1 and TMR2 is provided in Table 8.2 above.
Figure 8.1: Mathematics teachers’ instructional practices: Tasks
At the end of each code, for example TPK 1, there is a pair of numbers in parentheses
{4-1}. The 4 refers to the occurrence with which the code was attached to quotations
in the observation transcripts for a specific participant. This means that there were four
incidents during the two lessons observed from a specific participant where there was
evidence of the teacher building on the learner’s prior knowledge. Notice that at the
end of the three categories in Figure 8.1, namely: Tasks: Building on pre-knowledge
(TPK). Tasks: Modes of representation (TMR), and Tasks: Sequencing of content
21
(TSC), the numbers in parentheses are {0-2}, {0-3} and {0-3}. This indicates that the
codes TPK, TMR and TSC were not associated with quotations in the transcripts.
These are sub categories that were not coded as such. Instead their different lesson
dimension indicators were coded.
The code family created for the second category learning environments appears in
Figure 8.2 below. The broken line arrows again indicate the lesson dimension being
linked to the code family Instructional practices: Learning environment. The solid line
arrow with double equal signs indicates the code associated with the different lesson
dimensions.
Figure 8.2: Mathematics teachers’ instructional practices: Learning
environment
8.3.2.2. Inclusion criteria for coding the data
Table 8.2 indicates the inclusion criteria for coding the data. The table consists of the
different lesson dimensions, namely tasks and learning environment and the
respective lesson dimension indicators. The descriptions of the lesson dimension
indicators serve as inclusion criteria for coding the data from the observations.
Examples of each code are provided. These codes were used to analyse the raw data
and reporting of the data.
22
8.3.2.2. Exclusion criteria for coding the data
In the course of coding the observations, some of the events and dialogue were not
relevant and did not form part of my prescribed lesson indicators whereas others were
inaudible. These were excluded when the data were coded. In Table 8.3 below I listed
these exclusion criteria as well as examples of text that were excluded from coding.
Table 8.3: Exclusion criteria for coding of the data
Exclusion criteria Examples of text excluded from coding
Incidents during class observations when I could not hear what was said.
This occurred when the teacher attended to individual learners’ at their desks. Some of the data were inaudible when I did the transcribing.
Interruptions Teachers had to attend to people who knocked on the door or walked in.
8.4. Information regarding the two participants
In the section that follows I give some biographical information regarding the two
participants Elize and Alisha. I also provide some background information on the
school4 and the observed lessons. Pseudonyms were used to protect their identities.
8.4.1. The school
Due to the nature and small scale of this research both teachers were from the same
school. The school is a Section 21 (former model C) school in Pretoria. The language
policy of the school is Afrikaans and it accommodates a diversity of learners from all
cultures. The school is ’n quartile 5 school and has 1647 learners. Of these learners
91% are white, 6% are black and the rest are learners of other ethnic groups
represented in South Africa. The school has a teaching staff of 83 that are active in
classrooms.
4 Pseudonyms were used to protect their identities
23
8.4.2. Elize
Elize is 34 years old and completed a Bachelors of Commerce degree in financial
management in 2004 with Mathematics and Accounting as her major subjects. She
also completed her B.Com. Honours in 2009. She obtained both degrees from the
University of North West. She has eleven years’ teaching experience with two years’
experience specifically in Grade 9 mathematics teaching. Her Grade 9 mathematics
class consists of 31 learners and she teaches two subjects namely Mathematics as
well as Accounting.
The first lesson I observed were concerned with algebraic expressions as indicated in
the CAPS document under the content area of patterns, functions & algebra. This
lesson particularly focused on the multiplication of binomials. The second lesson I
observed was related to the drawing of linear graphs, which is still part of the same
content area. Elize was knowledgeable in her subject and her lessons included many
entertaining commentaries so that learners participated and enjoyed her classes.
Learners were involved by solving problems collaboratively in class and answering
questions.
8.4.3. Alisha
Alisha is 28 years old and completed her Bachelors of Education in the Further
education and training band with Mathematics and Mathematics didactics as two of
her major subjects. She also attended some courses on mathematics from the
Department of Education and the Society for Afrikaans Mathematics teachers. She
has six years’ experience of teaching Mathematics and it is her fourth year of teaching
Grade 9 specifically.
The first lesson I observed in her class were concerned with the drawing of linear
graphs that is part of the content area of patterns, functions & algebra as stated in the
CAPS documents. This lesson was an introduction to linear functions and how to draw
them on the Cartesian plane. The second lesson I observed a week later was intended
to be a summary of all the components and knowledge needed to draw linear
24
equations. This was particularly good as the researcher could follow the learners from
the introduction to where they would be able to draw linear equations on their own.
8.5. Lesson dimension 1: Tasks
In this section I will present the findings from the observations of Elize and Alisha. All
discussions on the sub-theme Tasks are organized according to the specific order of
the lesson dimension descriptors (codes) as it is indicated in Table 8.2 (Artzt, et al.,
2008). All the quotes from both teachers used in this section have been translated
from Afrikaans, as the school has a single medium language policy.
8.5.1. Elize’s instructional practices: Tasks
8.5.1.1. Tasks: Building on pre-knowledge
In the first lesson I observed, Elize started by marking homework given the previous
day. The marking of the homework which was multiplication of a monomials was used
as an introduction to the multiplication of binomials (TPK1). In her second lesson Elize
started the new topic, graphing of linear functions, with a revision of linear equations
and completing tables. This was done to show learners how to generate values for
variables 𝑥 and 𝑦 so that it may later be linked to co-ordinates for graphing on the
Cartesian plane (TPK1). Picture 8.1 below shows how Elize revised both the linear
equation with the table method in an attempt to link it with the concept of co-ordinates.
Picture 8.1: Illustrating linear equations and the table method.
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8.5.1.2. Tasks: Modes of representation
In both lessons observed, Elize facilitated content clarity to her learners by using
representations such as written work on transparencies, tables, symbols, formulas and
calculators (TMR1). In the first lesson Elize showed learners multiple strategies for
multiplication of binomials including the table format, using the so called FOIL (Fists,
Outsides, Insides, Last) strategy as well as multiplying all the values with each other
(TMR1). In Picture 8.2 an example is shown where Elize used the table method for
multiplication.
One formula that learners needed to use was y = mx + c. A learner also asked Elize
how she could use a calculator to determine the value of y for a given x value from the
table. Elize responded orally explaining to the learner how to insert brackets with the
given value from the table (TMR1). Another learner asked Elize to please explain why
one of the co-ordinates had a negative value. This question came from the following
examples calculating a value for the 𝒚 co-ordinate if x = 1 in y = 2x - 3:
y = 2(1) - 3 y = 2 - 3
y = -1
Picture 8.2: Illustrating the table method for multiplication.
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Elize went on explaining to the learner, which knew the substitution and multiplication
part, that to get to −1 he has to think moneywise. “If you only have a R2 coin but need
to pay someone R3 for their product, how much money do you still need?” The learner
replied that he still needs a R1. Whereby Elize replied “So needing or owing a R1 is a
negative thing as you cannot get the product that you want, thus we say it is -1”. This
was interesting for me as Elize drew on her accounting background knowledge to
explain a very difficult concept (TMR1). Elize generally drew the learners’ attention to
the required prior-knowledge needed for understanding the content of the specific
tasks (TMR2). To draw a linear equation on a Cartesian plane required values for
variables x and y.
She showed them how they could use their previous knowledge of products, equations
and tables to find values for these variables. To connect the learners’ prior knowledge
with the new knowledge she alternated between discussions with questioning and
classwork, of which the answers were discussed in class and self-assessed by
learners (TMR2). The picture below shows a class example used by Elize to link the
concepts of equation form, table and plotting a graph.
8.5.1.3. Tasks: Sequencing of content
Elize sequenced the tasks in both her lessons by progressing from easier to more
complex tasks (TSC1). She also sequenced her class activities: for example during
the first lesson on products she first checked homework and together with the class
she marked it. The homework progressed from introductory examples to more
complex tasks. The final questions as shown in Picture 8.4 below incorporated the
product of a quadratic function as well as the distributive law for a number in front of
the brackets.
27
Elize used the homework in both situations as a basis to build on what was done in
the lesson (TSC2). She appropriately applied the content of the work to more complex
situations, giving learners the ability to not only reinforce their prior knowledge, but
Picture 8.3: Illustrating how to plot a linear graph with co-ordinates.
Picture 8.4: Homework example incorporating all the elements of products
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also link it to their current situation (TSC2). In her interview Elize stated that she
develops her lesson and subsequent tasks to incorporate the learner’s prior
knowledge as introduction for new knowledge (TSC2). This was evident in her second
lesson where she used the learners’ prior knowledge of products, equations and tables
to build the idea of co-ordinates and the graphing of a linear function.
8.5.2. Alisha’s instructional practices: Tasks
8.5.2.1. Tasks: Building on pre-knowledge
In the first lesson I observed Alisha introduced a new topic. This was on the graphing
of a linear equation. Her introduction started with a PowerPoint slideshow listing all the
different ways of representing a relationship between two variables. Although an
attempt was made to link the new work with real life examples such as graphs from
the daily newspaper; she made no attempt to access learners’ prior knowledge of
products, equations or tables. The second lesson was a revision lesson that started
with the checking of learners’ homework, but no marking the homework at all. This
lesson followed the same trend as the first lesson. Learners were bombarded with a
list of steps and procedural mathematics to copy from the slideshow into their
textbooks bearing in mind that learners have access to these slideshows on their
tablets. In Pictures 8.5 and 8.6 respectively it shows the procedural steps (in bullet
form) that learners had to copy and follow.
Picture 8.5 and 8.6: Respectively illustrating the list of steps for the method of drawing linear
graph.
29
8.5.2.2. Tasks: Modes of representation
In both lessons observed, Alisha attempted to clarify the content to her learners by
using representations such as written work on the board and a PowerPoint slideshow
containing examples of tables and formulas (TMR1). She also encouraged learners to
make use of their calculators for finding the values of the variables x and y , however
she made no attempt to physically show learners how to do it (TMR1). She had the
technology to her disposal as the Casio calculator emulator was on her computer, but
failed to utilise it to connect to learner’s prior knowledge (TMR2). In another instance
Alisha attempted to clarify the use of a straight line graph by linking it to the Rand
Dollar exchange rate. This however was incorrect as any exchange rate graph will
never be a straight line because of currency fluctuations and validations occurring
almost every minute.
In the second lesson on the methods of drawing a linear graph the following was taught
within a single 35 minute period: 1) Using a table method for graphing co-ordinates by
selecting appropriate values that includes a zero, without explaining why the zero was
there; 2) Plotting of points (co-ordinates) on a Cartesian plane; 3) Connecting the
points to draw a graph; 4) The difference between a discrete and continues function;
5) Making use of the dual intercept method for graphing, that includes solving both
equations by making 𝑦 = 0 and 𝑥 = 0 to get the values of 𝑥 and 𝑦 and plotting both
intercepts on the axis as shown in Picture 8.7; 6) The difference between a positive
and negative gradient; 7) When the function was increasing or decreasing; and 8)
Showing learners how the graph of quadratic function would look like by drawing the
one in Picture 8.8 that was not according to standards.
30
Including all of the above, five minutes were also given to copy the slideshow from the
interactive board or the learners’ tablets into their workbooks. The learners complained
throughout the lesson that they did not understand the work (TMR1). Alisha used
technology together with these various representations in an attempt to have the
learners connect their prior knowledge with the new content, but the extent of the
content and the way she presented the work was too much for the learners to absorb
and led to learners’ confusion (TMR2). In many instances learners raised their hands
to ask questions and often times Alisha replied that that specific body of work was
dealt with in Grade 8 already, without giving a clear explanation. Some learners just
withdrew from all activities during the lesson and were looking out the window (TMR2).
8.5.2.2. Tasks: Sequencing of content
Alisha did not give much attention to the sequencing of tasks in order for the learners
to obtain cumulative understanding of the content (TSC1). In her first lesson the table
method was introduced, but no logical connection to products or solving equations
was established (TSC1). This was followed by her first example being that of y = 2x + 3
to draw a linear graph. She then moved directly into the negative graph of y = -2x + 1
Picture 8.7: Illustrating the dual intercept Picture 8.8: Illustrating a graph of a quadratic function
31
in her second example (TSC1). When learners asked why the graph was now in the
opposite direction she replied: “We will get to that part later, for now all you need to
know is how to graph a function” (TSC1). In both lessons that were observed, Alisha
used a PowerPoint slideshow to show tables that were already completed with values.
She thus did not give learners a chance to draw and complete a table on their own so
that they could revise prior work, but also reinforce the new concept of co-ordinates
(TSC2). In the second lesson Alisha gave homework on the graphing of a linear
function using the dual intercept method. This was suitable for only one aspect of the
current work covered in the lesson. There was no evidence of different examples that
could have incorporated both the table as well as the dual intercept method to help
identify learner misconceptions (TSC2).
8.6. Lesson dimension 2: Learning environment
Presented in this section is the findings from the observations of Elize and Alisha
related to the subtheme of learning environment, as indicated in Table 8.2 (Artzt, et
al., 2008). All the quotes have been translated from Afrikaans as mentioned
previously.
8.6.1. Elize’s instructional practices: Learning environment
Elize’s teaching style varied between being a facilitator and mediator of learning
(LEIP1). She proficiently used instructional strategies such as class discussions and
direct instruction (LEIP1). The use of these strategies provided opportunities for the
involvement of the learners and facilitated goal attainment (LEIP1). However in the
second lesson Elize could not finish the final example and homework was handed to
learners in a rush because of poor pacing of the lesson.
8.6.2. Alisha’s instructional practices: Learning environment
Alisha used direct instruction (lecturing), a teacher-centred approach as well as a little
discussion with a few learners in front of the class (LEIP1). On one occasion in the
first lesson learners worked on the board. All three learners showed three different co-
ordinates on the Cartesian plane so that Alisha could plot the points and draw the
32
graph (LEIP1). The direct instruction strategy Alisha used, did not always support
learner involvement and goal attainment and she was not aware of the learners’ lack
of knowledge and skills regarding the topic she covered. Alisha assumed that if she
understood the work, the learners would understand it too (LEIP1). No assessment of
the learners’ knowledge was done and there was no evidence that Alisha’s goals had
been reached (LEIP1).
Table 8.4: Summary of both teacher’s instructional practices and learning
environments
LESSON
DIMENSIONS
DESCRIPTION OF LESSON DIMENSION INDICATORS
Tasks:
Elize
Alisha
Building on prior knowledge (TPK1)
In both lessons Elize accessed her learner’s prior knowledge and used it as a platform to build on new knowledge. She attempted to move from the known to the unknown.
Alisha seldom accessed her learner’s prior knowledge. Although she made attempts to do so in her slideshow, her teacher-centred approach left little room for discussions so that learners could access their prior knowledge.
Modes of representation
Elize made use of representations such as written work on transparencies and calculators These representations allowed her to link learners’ prior knowledge with the new content of the day.
Alisha used representations such as a slideshow, written examples on the board, symbols, formulae, tables, graphs and calculators. She could not proficiently use the various representations to connect learners’ prior knowledge with the new mathematical situation.
Sequencing and difficulty levels
The given tasks were sequenced over the different lessons and were Appropriate.
The tasks chosen were appropriate and on Grade 9 level, but were not presented logically or in context to ensure that learners were motivated.
33
Modes of strategies and pacing
She used instructional strategies such as discussions and direct instruction. The use of discussions provided opportunities for the involvement of learners and facilitated goal attainment. Learners had enough time to express themselves and explore their ideas and solutions and there was a logical flow in her lessons.
Her teaching was that of an authority style. She used direct instruction (lecturing) as instructional strategy and only once allowed three learners to work on the board. Typical of teacher directed lessons, learners were involved copying work from the board, listening to explanations of the teacher and answering basic low level questions.
8.7. Findings from interviews
8.7.1. Elize’s interview
In an interview, after the two observed lessons, Elize explained that she always
attempts to access her learner’s prior knowledge in all her lessons (TPK1). “I believe
in moving from the known to the unknown. Building on what learners know already.
This is how my children really learn.” Elize believes that her tasks are sequenced to
attain this goal i.e. moving from the known to the unknown (TSC1). Elize believes that
with time, every teacher develops the ability to assess their learners. This enables her
to determine what tasks are suitable during instruction and which ones are best for
homework (TSC2). She also indicated that she would more resources and training to
incorporate technology into her classroom presentations (TMR1). “Learners need to
be able to connect their classroom mathematics with the outside world and I think this
can be done with technology like internet in my class (TMR2). Elize explained that her
school is in the process of upgrading classrooms to become more technologically
friendly. It is however expensive and her class would be on the next phase of
installations, but she also stated that incorporating technology into teaching must be
accompanied by quality teacher training and that she would have to attend courses to
improve her understanding of educational technology.
When asked how she feels about different teaching strategies, Elize indicated that she
enjoys co-operative learning and wants all her learners to participate in whole class
discussions (LEIP1); but with such a full CAPS curriculum and discipline also being a
34
problem, she often reverts to direct instruction (LEIP1). Barriers which Elize identified,
was the big administration load placed on teachers and also the intercom system in
the classroom that often disrupts her lessons.
8.7.2. Alisha’s interview
In Alisha’s interview when asked what she would change in her practise, she indicated
that she would want to incorporate even more multimedia in her classroom (TMR1).
She also believes that learning takes place trough interactive lessons. “I believe that
learners need to active in the classroom i.e. they must participate. Doing multiple
questionnaires on their tablets is something that I am incorporating now” (TMR1
&LEIP1). When asked how she develops her lessons, Alisha replied that she starts
with an introduction, then remind them (her learners) of previous lessons and let the
lesson flow into the new work (TSC1). Alisha also indicated on the topic of sequencing
that she strongly believes in three steps to teaching. Firstly teach the theory, secondly
do enough examples and lastly practise at home (TSC2). Barriers Alisha identified in
her teaching of mathematics were: 1) the time limit according to CAPS for certain
topics which in practice is not sufficient time to cover the topic and 2) learner motivation
towards Mathematics as a subject being very low.
8.8. Conclusion
In this chapter, I discussed the data collection process that took place at a school in
Pretoria. Data were collected from the two participants by means of two lesson
observations with an interview conducted after the second lesson. As previously
mentioned, I adopted a deductive approach to coding the data as I had identified two
themes under Mathematics teachers’ instructional practices namely tasks and the
learning environment. Different categories for each theme were chosen according to
the work of Artzt et al. (2008). In this chapter I also presented the data and findings of
the two participants.
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9. Conclusions and Implications
9.1. Introduction
In this section I provide a short summary of the previous sections up to this point,
answer the research questions that guided this study and also reflect on my research
as to what I would have done differently. Furthermore this is followed by the
conclusions, recommendations and limitations of my study.
9.2. Summary of the sections
In sections 1 to 5 I introduced and contextualised the research study. The purpose of
this research was to investigate, by means of a case study, the way in which
Mathematics is taught to Grade 9 learners. This was done with the view to determining
the teachers’ PCK as it is reflected in their instructional practices. I also deliberated
the state of Mathematics teaching in South Africa, drawing on the results of relevant
tests like TIMMS and the ANA. The problem statement and the rationale of my study
are also discussed followed by the research questions.
Section 6 presented an in-depth analysis of the findings in the relevant literature as
well as the conceptual framework on which the study is based. I discussed the demand
for the subject Mathematics within the South African school system as well as the
increased demand for quality mathematics teachers. Following this was a discussion
of the meaning of teachers’ instructional practices and the value of various approaches
to teaching. Attention was given to the different domains of teachers’ knowledge and
their instructional practices. The conceptual framework, which is based on concepts
and theories from relevant work in the literature, was then discussed.
A description of the qualitative methodology used in this study was reported in Section
7. I discussed social constructivism as my research paradigm, and the nature of my
study as subjective and interpretive. This is an exploratory case study. Observations
were used to examine teachers’ instructional practices and to study demonstrations of
their PCK. Interviews were used to determine why teachers do what they do in class
and to determine how they apply their PCK during their instructional practice. ATLAS.ti
6.2 was used to analyse the video data. I lastly discussed the trustworthiness of the
study and the ethical considerations that were taken into consideration.
36
In section 8 I briefly reported on the data collection process and presented the findings.
A DEDUCTIVE-inductive (uppercase denotes the preference given to the style of
analysis) approach to coding the data was used as I identified two lesson dimension:
Tasks and Learning environment. After this deductive phase of analysis, inductive
analysis was done when I studied the organised data in order to explore new patterns
and trends. I presented the findings from the data obtained through class observations
and interviews according to the different categories provided in Table 8.2.
9.3. The research questions
Based on the rational that a teacher not only needs high-quality content knowledge,
but also a high level of pedagogical content knowledge to assure effective teaching of
mathematics. I decided to explore Grade 9 Mathematics teachers’ PCK, regarding
teaching as reflected in their instructional practices. In order to do so, the following
main research question was formulated: How can Grade 9 Mathematics teachers’
PCK regarding teaching as reflected in their practices be described? To address this
main question, the following four sub-questions guided the enquiry:
1. To what extent did the teachers base new knowledge on the learners’ prior
knowledge?
2. Which forms of representation did the teachers use?
3. How can the sequencing of content by the teachers during the lesson be
described?
4. How appropriate were the teaching strategies used by the teachers?
I used an adapted version of the theoretical framework provided by Artzt et al. (2008)
on teachers’ instructional practices to contextualise and interpret my results. To
answer the above questions, the participants’ instructional practices were described
according to the lesson dimensions as indicated in this study’s conceptual framework.
37
9.3.1. To what extent did the teachers base new knowledge on the learners’
prior knowledge?
The novice teacher in my study
Alisha attempted to no extend to access her learner’s prior knowledge. There was no
clear indication of her attentions to link new knowledge to that of her learner’s prior
knowledge. In both her lessons she never referred back to previous mathematical
concepts that were already dealt with in the curriculum. For instance, in her attempt to
explain the graph of a linear function to her Grade 9 learners, there was no evidence
of revisiting or even mentioning the link between solving linear equations using tables
etc. and now graphing them.
The expert teacher in my study
Elize had a clear understanding of what her learners knew and where she wanted
them to be in terms of knowledge by the end of each period. In both her lessons she
linked the learner’s prior knowledge to a big extend with the new knowledge presented.
In her explanation of the linear graph Elize started with simple examples just generate
values for variables x and y so that it may later be linked to co-ordinates for graphing
on the Cartesian plane
In this research, the novice teacher (Alisha), unlike experienced teacher (Elize) did to
no extent determined nor used her learners’ prior knowledge to facilitate the
assimilation of new content knowledge. This finding strongly confirms Sidiropolous’
(2008) finding that one of the two teachers in her research group did not determine his
learners’ prior knowledge or use his learners’ prior knowledge to facilitate the
assimilation of new content knowledge. Carpenter, et al. (1988) also agrees that
novice teachers tends to make broad pedagogical decisions without assessing
students' prior knowledge, ability levels, or learning strategies.
9.3.2. Which forms of representation did the teachers use?
The novice teacher in my study
Alisha used multiple representations such as a slideshow, written examples on the
board, symbols, formulae, tables, graphs and calculators. From the observations one
38
could conclude that Alisha spends time in gathering and developing technological
resources for her lessons. However multiple resources does not constitute for learner
understanding alone. Alisha could not proficiently use the various representations to
connect learners’ prior knowledge with the new mathematical situation thus leaving
many learners not being able to understand and seemed uninterested.
The expert teacher in my study
Elize made use of representations such as written work on transparencies and
calculators. Although Elize was not as technologically savvy what she did very well
with her learners was keeping them interested by linking their prior knowledge with the
new content of the day and using real life examples to generate discussions. Her
learners actively engaged in the class and was interested in learning new concepts.
This revealed that high-quality subject knowledge needed to be supplemented by
knowledge of the learners and general knowledge of the real world, in order to
generate meaningful discussions and keep learners actively involved.
In this study both participants used various representations during their classes as
was also found with all the participants in the other research studies (Sidiropolous,
2008; Venkat & Graven, 2008; Venkat, 2010). Elize had also expressed a need for
more resources and adequate training to increase the use of technology in her
Mathematics classroom. This finding correlates with that of Landry (2010) were
several participants identified more resources and training to assist them in being more
effective at teaching mathematics using technology.
9.3.3. How can the sequencing of content by the teachers during the lesson be
described?
The novice teacher in my study
The tasks that Alisha chose were appropriate and on Grade 9 level, but were not
presented logically or in context to ensure that learners were motivated. According to
Artzt et al. (2008) meaningful tasks can provide opportunities for learners to connect
their knowledge to new information and to build on their knowledge and interest
39
through active engagement. Alisha’s sequence of content could thus be described as
not meaningful as it was based on content only, not incorporating learners’ prior
knowledge or providing any opportunity for learners to actively engage.
The expert teacher in my study
Elize chose her content and tasks meaningfully. This correlates to Artzt et al. (2008)
as he explains that meaningful tasks can provide opportunities for learners to connect
their knowledge to new information and to build on their knowledge and interest
through active engagement. In her interview she explained that she believes in moving
from the known to the unknown. Thus she developed her lessons and tasks to help
her learners move progressively, building on prior knowledge, setting the stage for
future work and at the same time giving ample practise to reinforce the necessary
content. This was also evident from the observations, as Elize started her class with
questions and marking of previous homework. Both her class examples and
homework given incorporated previous knowledge before moving to new knowledge.
9.3.4. How appropriate were the teaching strategies used by the teachers?
The novice teacher in my study
Alisha’s teaching strategies could be described as not appropriate as it was of an
authoritative style. She used direct instruction (lecturing) mode as instructional
strategy. Her lessons were teacher-centred as she believed that her role as teacher
was to transmit mathematical content, demonstrate procedures for solving problems
and explain the process of solving sample problems. However, Artzt et al. (2008)
suggests that this approach is not ideal as the teacher-centred approach can serve as
a mask for teachers who do not fully understand the content, the learners or the
pedagogy, as was found in her practices.
The expert teacher in my study
Elize used instructional strategies that was more appropriate as it included discussions
and direct instruction. Her teaching strategy could be described as a combination of
teacher- and learner-centred as she started both observation lesson with probing
question to involve learners in providing answers. According to Artzt, et al.(2008) a
40
learner-centred approach to teaching requires the teacher to create opportunities for
learners to achieve understanding through active engagement with each other and the
problem-solving process. Elize facilitated whole class discussion that learners actively
engaged in.
9.4. Summary of my findings
In summary: My study seems to provide evidence, from both teachers’ instructional
practises, that the experienced teacher’s (11 years’ experience) PCK could be
described as sufficient whilst the novice teacher (4 years’ experience) only had
superficial PCK. My finding that the experienced teachers had developed PCK
confirms the findings of Ball (1988), Ball et al. (2005), Koellner et al. (2007), Ma (1999)
and Shulman (1986) that PCK can be developed only over time through experience in
the classroom.
According to my findings, it also seems that a teacher’s PCK may play a significant
role in the variety and quality of their instructional practices. Elize, with 11 years’
experience of teaching Mathematics and sufficient PCK, had used a combination of
teacher- and learner-centred approach compared to Alisha (with 4 years teaching
experience and superficial PCK) which included only a teacher centred, lecture style
approach.
9.5. What would I have done differently?
During the write-up stage, I realised that I had missed valuable dialect between the
teacher and the learners at their desks as I did not want to impose by moving around
in class with a video camera. More information regarding the teachers’ PCK would
possibly have emerged from this discourse.
9.6. Limitations of the study
Due to the small nature of this study, data were gathered from a very small number of
Mathematics teachers and generalization of the results is impossible. Another
limitation is the fact that only two observations per teacher were carried out and they
were all done in the second part of Term 3. I am also acutely aware that different
41
researchers may interpret my data differently. My own perspective is bound by space,
time and personal experience.
9.7. Conclusion Some conclusions with regards to a Grade 9 Mathematics teachers’ PCK as reflected
in their instructional practices appear below.
Grade 9 Mathematics teachers’ instructional practices should be predominantly
learner-centered, including the use of active learning instructional strategies
such as cooperative learning and discussions.
Grade 9 Mathematics teachers need ample time for building PCK through
hands-on experience in class. Initial teacher training programs must take this
into consideration.
9.8. Possible implications of the findings
This research indicates that instructional practises are linked to PCK and the
development thereof. The instructional practices and teaching strategy of Alice, the
novice teacher, proved to be unproductive resulting in discouraged and uninvolved
learners. Elize on the other hand who had more years teaching experience, was
knowledgeable and competent and illustrated more productive teaching. Initial teacher
training programs must therefore focus on giving pre-service teacher sufficient
classroom experience so that they can develop their PCK. This finding is supported
by Ball (1988), Koellner et al. (2007), Ma (1999) and Shulman (1986) were they
conclude that PCK can only be developed over time through experience in the
classroom or practice and can therefore not be taught. Effective and purposeful
training of pre- and in-service Mathematics teachers is of utmost importance in South
Africa, a finding that was also reported by Sidiropolous (2008).
Some key factors that should be part of all Mathematics teacher’s instructional
practises are:
Using a learner-centred approach and appropriate instructional strategies.
Engaging learners in discussions thereby enabling them to communicate their
thinking through the use of appropriate terminology.
42
Using various instructional resources to connect learners’ knowledge with new
situations.
These key factors were also confirmed by Artzt et al. (2008).
9.9. Recommendations for future study
From this study several aspects of the teaching and learning of Grade 9 Mathematics
require further research. These include investigation into:
The knowledge required to engage learners in such a manner as to explore the
depths of their prior knowledge during teaching.
Identification of authentic and relevant 21st Century technologies that can be
incorporated to enhance Mathematic teachers’ PCK through quality
collaborative platforms.
Identification of appropriate technologies that would improve a Mathematics
teacher’s instructional practises.
43
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Appendices Appendix A Letter of consent to the Mathematics learners
Appendix B Letter of consent to the principal
Appendix C Letter of consent to teachers
Appendix D Ethical clearance certificate
Appendix E Observation sheet for observing Mathematics teachers’ lessons
Appendix F Interview schedule
49
Appendix A: Letter of assent to the learners FACULTY OF EDUCATION
FACULTY OF EDUCATION Dr. J.J. Botha Natural Science Building 4-13 Groenkloof campus, UP
e-mail: [email protected] Tel: 012 420 5623
1 July 2015
Dear ………………..………………
Letter of assent to the learners
You are invited to participate in a research project aimed at investigating teachers’ instructional
practices. This research will be reported upon in my honours research report conducted at the University
of Pretoria under the supervision of Dr JJ Botha. Your parents also have to grant consent for your
voluntary participation, but you should also declare yourself willing to participate in this study.
Your participation in this research project is voluntary and confidential. You will remain anonymous and
will participate as usual during the lessons and you will have no contact with me. I would like to observe
your mathematics teacher during the third term of this year while teaching mathematics to your class.
Two observations will be done and it will not interrupt your school timetable or your subject progress.
The lessons will be audio recorded.
You may decide to withdraw at any stage should you not wish to continue with your participation. Your
decision to accept/decline involvement in this research will not affect your studies. If you are willing to
participate in this project, please sign this letter as a declaration of your assent, i.e. that you participate
in this project willingly by allowing me to observe your Mathematics teacher and that you understand
you may withdraw from the research project at any time. For any query or more clarification please
contact Dr Botha on the cell phone number provided above.
Yours sincerely …………………………………………………………. Date:……………………………… Co-researcher: …………………………………………………………. Date:…………………………… Researcher: Dr. J.J. Botha
50
…………………………………………………………. Date:……………………………… Acting HOD: Dr. L.S. van Putten _________________________________________________________________________________ I the undersigned, hereby grant assent to the co-researcher to observe my Mathematics teacher when presenting our Mathematics classes as part of a research programme. Learner’s name: ……………………………………………………………………………..…….. Learner’s signature: ........................................................... Date:………………………………
51
Appendix B: Letter of consent to the Principal
FACULTY OF EDUCATION
Dr. J.J. Botha Natural Science Building 4-13 Groenkloof campus, UP e-mail: [email protected]
Tel: 012 420 5623 1 July 2015 Dear Dr/ Mrs/ Mr ………………..………………
Letter of consent to the Principal
I hereby request permission to use your school in a research project aimed at investigating the relationship between Mathematics and Mathematical Literacy teachers’ mathematical content knowledge and their instructional practices. This research project is funded by the Research Development Programme (RDP) and conducted by Dr JJ Botha at the University of Pretoria. My research project forms part of Dr Botha’s project and is entitled: Mathematics teachers’ pedagogical content knowledge regarding teaching as reflected in their practices. I would like to invite a Senior phase Mathematics teacher to participate in my research project. The teacher’s participation in this research project is voluntary and confidential. It is proposed that the teacher forms part of this project’s data collection phase by being observed two times when teaching one of the Senior phase classes and being individually interviewed once afterwards. The observations will be done at a time convenient to you and should not disrupt your timetable. The interview should be conducted at a time and place convenient to you and should not take longer than 45 minutes. The lessons and the interviews will be audio-taped by me in order to have a clear and accurate record of all the communication that took place. Only audio-recordings will be done so that minimal disruption for the learners is assured. During my observation of the lessons, I will make field notes on an observation sheet that has been prepared in advance based on the research questions to be answered. Should you declare yourself willing to participate in this study, confidentiality and anonymity will be guaranteed at all times. You may decide to withdraw at any stage should you not wish to continue with your participation. Your decision to accept involvement in this research project will hopefully contribute to the improvement of Mathematics teachers’ instructional practices. If you are willing to allow a member of your staff to participate in this project, please sign this letter as a declaration of your consent. Yours sincerely …………………………………………………………. ........... Date:……………………………… Co-researcher:
…………………………………………………………. Date: ………………………………
Researcher: Dr. J.J. Botha
52
…………………………………………………………. Date: ………………………………
Acting HOD: Dr. L.S. van Putten
_________________________________________________________________________________
I the undersigned, hereby grant consent to the co-researcher to observe a Senior phase
Mathematics teacher’s classes and conduct an interview with the teacher as part of a research
programme.
School principal’s name ……………………………………………………………………………..…….. School principal’s signature ............................................... Date:……………………………… E-mail address ………………………………………………. Contact number …………………….
53
Appendix C: Letter of consent to the parents
FACULTY OF EDUCATION
Dr. J.J. Botha Natural Science Building 4-13
Groenkloof campus, UP e-mail: [email protected]
Tel: 012 420 5623 1 July 2015
Dear Dr/ Mrs/ Mr ………………..………………
Letter of consent to the parent(s)/guardian(s)
Dear parent(s)/guardian(s) REQUEST TO ALLOW YOUR CHILD TO PARTICIPATE IN A RESEARCH PROJECT
I am enrolled for my Honour’s degree at the University of Pretoria at the Department of Science,
Mathematics and Technology Education, under the supervision of Dr JJ Botha. I hereby request you to
grant permission for your child to participate in my research project.
The topic of my research project is: Mathematics teachers’ pedagogical content knowledge regarding
teaching as reflected in their practices. The aim of this study is to determine how teachers plan their
lessons, but also to observe the teaching strategies used when presenting their Mathematics lessons.
The findings of this study may contribute to teachers enhancing their practices to ensure learners’
developing mathematical understanding in order to contribute and understand the world they are living
in.
To collect my data, I have to observe your child’s Mathematics teacher teaching two different lessons.
I want to emphasise that my observations will not interrupt the school timetable or the mathematics
classes. I will not have any direct contact with the learners, as the focus is on the teacher’s instruction
in class. The lessons will however be audio-recorded, but the audio recordings will only be used by me
and should my supervisor want to listen in order to assist with the analysis of the data. All learners will
remain anonymous and no names will be revealed. Learners’ participation in this study is voluntary and
they may withdraw from the study at any time without any consequences. Should your child want to
withdraw from the study, I will just position myself in class in such a way that your child will sit behind
me and will his or her contributions in class not be considered in any way as part of the data being
collected.
54
Yours sincerely …………………………………………………………. ........... Date:……………………………… Co-researcher: …………………………………………………………. ........... Date:………………………………. Researcher: Dr. J.J. Botha …………………………………………………………. ....... Date:……………………………… Acting HOD: Dr. L.S. van Putten _________________________________________________________________________________ I, ______________________________________________________ hereby give permission for my
child, _______________________________________________________, to participate in this
research study, by allowing the co-researcher to observe my child’s Mathematics teacher and to also
make audio recordings of the two lessons. I am aware that my child will remain anonymous and that
the findings of this research will be used to promote teaching and learning.
Signed:______________________________________
Date:__________________________
55
Appendix D: Ethical clearance certificate
Ethics Committee 5 August 2015
Dear Dr. Botha, REFERENCE: UP 14/04/01 We received proof that you have met the conditions outlined. Your application is thus approved, and you may continue with your fieldwork. Should any changes to the study occur after approval was given, it is your responsibility to notify the Ethics Committee immediately. Please note that this is not a clearance certificate. Upon completion of your research you need to submit the following documentation to the Ethics Committee: 1. Integrated Declarations form that you adhered to conditions stipulated in this letter – Form D08 Please Note:
Any amendments to this approved protocol need to be submitted to the Ethics Committee for review prior to data collection. Non-compliance implies that the Committee’s approval is null and void.
Final data collection protocols and supporting evidence (e.g.: questionnaires, interview schedules, observation schedules) have to be submitted to the Ethics Committee before they are used for data collection.
Should your research be conducted in schools, please note that you have to submit proof of how you adhered to the Department of Basic Education (DBE) policy for research.
Please note that you need to keep to the protocol you were granted approval on – should your research project be amended, you will need to submit the amendments for review.
The Ethics Committee of the Faculty of Education does not accept any liability for research misconduct, of whatsoever nature, committed by the researcher(s) in the implementation of the approved protocol.
On receipt of the above-mentioned documents you will be issued a clearance certificate. Please quote the reference number: UP 14/04/01 in any communication with the Ethics Committee.
Best wishes, Prof Liesel Ebersöhn Chair: Ethics Committee
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Faculty of Education
Appendix E: Observation sheet for observing Mathematics teachers’ instructional practices
OBSERVATION SHEET (To be used for both observations per teacher)
Name of school
Name of co-researcher
Subject observed Mathematics
Grade observed
Number of learners in class list (present in class)
Topic of the lesson
Name of teacher
Date of observation
Observation number
Table 1 below provides the Description of the codes, while Table 2 is an open observation sheet where field notes can be made.
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ASSESSING TEACHERS’ INSTRUCTIONAL PRACTICES THROUGH OBSERVATIONS (Audio-record lessons and make field notes during observations) Table 1: Description of codes
LESSON DIMENSIONS SCALE DESCRRIPTION OF CODES
Tasks
Building on pre-knowledge TPK1. There should be a logical flow in the lesson such as revising prior knowledge before introducing
new content.
Modes of representation
TMR1. Uses representations such as oral or written language, symbols, diagrams, graphs, tables, manipulatives, and computer or calculator representations to accurately facilitate content clarity.
TMR2. Provides multiple representations that enable learners to connect their prior knowledge and skills to the new mathematical situation such as graphs, tables, formulae.
Sequencing of content
TSC1. Sequences tasks and learning activities so that learners can progress in their cumulative understanding of a particular content area and can make connections between ideas learned in the past and those they will learn in the future such as working from easy to difficult and known to unknown tasks.
TSC2. Uses tasks, including homework that is suitable to what the learners already know and can do and what they need to learn or improve on. Tasks should involve past work, reinforce current work and set the stage for future work such as tasks where opportunity is given to practice identified or predicted learners’ misunderstandings.
Learning environments
Modes of instruction/pacing
LEIP1. Uses various instructional strategies that encourage and support student involvement as well as facilitate goal attainment such as cooperative learning, learners explaining work at the board, direct instruction (lecturing), abstract procedural, group work, active learning, discussion, problem solving, inquiry and team-teaching.
Evaluation scale: Description of the scale. 3 = commendable (strong presence of indicator); 2 = satisfactory (indicator is somewhat present); 1 = needs attention (there is very little presence of indicator); N/O = not observed or not applicable.
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Table 2: Researcher’s comments
LESSON DIMENSIONS SCALE COMMENTS (support with examples)
Tasks
Building on pre-knowledge
TPK1.
Modes of representation
TMR1. TMR2.
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Sequencing of content
TSC1. TSC2.
Learning environments
Modes of instruction/pacing
LEIP1.
Evaluation scale: Description of the scale. 3 = commendable (strong presence of indicator); 2 = satisfactory (indicator is somewhat present); 1 = needs attention (there is very little presence of indicator); N/O = not observed or not applicable.
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Appendix F: Interview schedule (Conducted after the observations)
INTERVIEW SCHEDULE
Semi-structured interview
GENERAL INFORMATION
Name of school
Name of researcher
Name of teacher
Date of interview
Teacher’s qualification
Level of Mathematics education
Number of years teaching Mathematics
Number of years teaching the specific grade
Courses attended on teaching Mathematics
This interview consists of two sections and gives the teachers an opportunity to reflect on their
practices. The interview is a discussion on teachers’ planning of their lessons and how the teachers
experience their instructional practices.
1. What do you want to change in your practice? How will you change it?
2. How does learning take place?
3. Tell me how do you plan the development of your lessons?
4. How do you choose which tasks the learners should do in class and which they should do at home?
5. How do you feel about creating opportunities for learners to discuss the work with you, but also
with other learners during the lesson?
6. Do you plan the oral questions prior to the presentation of your lesson? If so, how do you plan
the questions as part of the development of the lesson?
7. Which teaching strategies do you value important in teaching Mathematics? When teaching the
specific topic you did? Why? (Give prompts if required)
8. What barriers do you experience that keep you from having the ideal classroom situation?