CHAPTER THREE RESEARCH DESIGN AND METHODOLOGY
The preceding chapter presented the theoretical underpinnings of this enquiry. In order to fully address the study objectives, this chapter starts with a synopsis of the research paradigm and research design, and then it provides a description of the research site and participants. In addition, instruments, data collection and analysis procedures, validity and reliability, and ethical issues will be discussed.
The major question that guided the study was, “To what extent does teaching and learning of transformation geometry utilise students’ lived experiences?” This research study has been undertaken primarily to identify and suggest how Transformation Geometry thinking can be enhanced in learners through incorporating their real-life experiences. The study is significant in that incorporating students’ lived experiences in transformation geometry classes is bound to help teachers deliver lessons in the subject topic in a way that will excite students, assist their connection and application of « real world » settings to the concepts and extend students’ abilities to solve mathematics problems in other context (Dickson et al., 2011). For that purpose, it was important that the selected research design and methods be relevant and appropriate in answering the research questions. The following are the research questions that guided the study:
1. What are teachers’ perceptions about the mathematics involving transformation geometry concepts contained in the students’ out-of-school activities?
2. How is the context of transformation geometry teaching implemented by practising teachers in Zimbabwean rural secondary schools?
3. To what extent are students’ out-of-school experiences incorporated in transformation geometry tasks?
4. How is transformation geometry, as reflected in official textbooks and suggested teaching models, linked to students’ out-of-school experiences?
Denzin and Lincoln (2000:157) define a research paradigm as, “a basic set of beliefs that guide action, dealing with first principles, ‘ultimate’s or the researcher’s worldviews”. In other words, a paradigm is an action of submitting to a view. In this study it was important to define the researcher’s paradigm choice as it served to guide in exploring the extent to which teachers use learners’ real-life experiences to enhance learners’ transformation geometric thinking.
This study is oriented in the interpretive research paradigm. The interpretive paradigm holds the view that people have reasons why they act the way they do, and that to understand the reasons behind human action requires not detachment from, but rather direct interaction with the people concerned (Connole, 1998; Schwandt, 2000). Like other research paradigms, the interpretive paradigm is characterised by its own ontology, epistemology and methodology (Terre Blanche & Kelly, 1999).
The interpretive tradition assumes that people’s subjective experiences are real and should not be overlooked (ontology), that these experiences can be understood by interacting with the people concerned and listening to what they have to say (epistemology), and that qualitative research techniques are best suited to gaining an understanding of the subjective experiences of others (methodology) (Blanche & Kelly, 1999). Ontology defines the nature of reality that is to be studied and what can be known about it; epistemology defines the nature of the relationship between the researcher (knower) and what can be known; and “methodology specifies how the researcher may go about practically studying what can be known” (Blanche & Durrheim, 1999:6).
The epistemological position regarding the current study was formulated as follows:
a) data are contained within the perspectives of people that are involved with the teaching (teachers of mathematics) and how students learn the geometric transformations in rural secondary schools in the district of Mberengwa, Zimbabwe, through observations and interrogation; and
b) to be engaged with the participants (teachers of mathematics and ordinary level students of mathematics) in collecting the data.
Based on the qualitative research paradigm I took the transcendental phenomenology approach by Moustakas (1994), adapted from Husserl (1931) to generate an essence of the lived experience of participants. The general purpose of the phenomenological study is to understand and describe a specific phenomenon in- depth and reach at the essence of participants’ lived experience of the phenomenon (Yuksel & Yildirim, 2015). The intention of the study therefore was to explore Mathematics teachers’ use of students’ out-of-school experiences in the teaching and learning of Geometric Transformations and the associated benefits, through use of different data collection techniques such as lesson observations and post lesson interviews (Yin, 2003: Creswell, 2013).
In this study, the researcher was able to see and hear mathematics teacher practices with transformation geometry. The use of this design ensured the researcher to arrive at answers to questions “What, how and to what extent” teachers use students’ real life experiences in teaching transformation geometry concepts (Creswell, 2013).
In this study, the object of the phenomena is use of real life experience in the teaching of Transformation Geometry in Mathematics classes. The subject is teachers of mathematics. Therefore, the study explored how teachers use learners’ real-life experience in Transformation Geometry (T.G) Mathematics classes. In this study, the act of experience which is the meaning of the essence occurred after the imagination variation is using real life in Transformation geometry (T.G) teaching in the classroom (Yuksel & Yildirim, 2015).
Structure of study design
In order to understand the phenomenological idea, the following key concepts of the phenomenology philosophy are examined: lived experience, intentionality, noema-noesis, epoché and co-researchers.
Phenomenological research investigates the lived experience of participants with a phenomenon. Phenomenological studies start and stop with lived experience which should be meaningful and significant experience of the phenomenon (Creswell 2007; Moustakas, 1994, Thani, 2012; van Manen, 1990).
In this study, the researcher was interested directly with related lived experiences of the phenomenon, that is, teachers’ use of real life experiences of the learner. Therefore, participants demonstrated some meaningful and significant experience of how they use learner experiences in Transformation Geometry teaching.
Husserl (1970) argues that there is a positive relationship between perception and objects. The object of the experiences is actively created by human consciousness as we always use our consciousness in thinking. It needs perceiving or conceiving an object or an event (Holstein & Gubrium, 2000). Therefore, for Husserl (1931), intentionality is one of the fundamental characteristics of the phenomenology that is directly related to the consciousness.
Intentionality refers to doing something deliberate. For example, in this study, using learners’ real-life experience for enhancing Teaching and learning in TG is an intentional experience of teachers’ non-mental activities (Yuksel & Yildirim, 2015). Teachers’ examples of TG concepts in their teaching are intentional acts dependent on teachers’ consciousness. Therefore, the act of experience is related to the meaning of a phenomenon. Thus, the essence of the phenomenon is derived from the act of teachers experiencing perceived real-life examples of TG concepts in the classroom. This means that “the object exists in the mind in an intentional way” (Kolkelman, 1967; Moustakas, 1994:28). Therefore, intentionality reflects the relationship between the object and the appearance of the object in one’s consciousness.
In the transcendental phenomenology design, the intentionality has two dimensions, noema and noesis (Yuksel & Yildirim, 2015). Noema is the object of experience or action, reflecting the perceptions and feelings, thoughts and memories, and judgments regarding the object. Noesis is the act of experience, such as perceiving, feeling, thinking, remembering, or judging (Yuksel & Yildirim, 2015). The act of experience is related to the meaning of a phenomenon. In this study, while real life learner experiences related to TG concepts is the noema of the experience, using the real-life experiences for purposes of teaching concepts is the noesis of the experiences. Noema and noesis are interrelated and cannot exist independently or be studied without the other (Cilesiz, 2010).
Epoché is a Greek word used by Husserl (1931) meaning to stay away or abstain from presupposition or judgments about the phenomena under the investigation (Langdridge, 2007). Basically, Epoché allows the researcher to be bias-free to describe the reality from an objective perspective. For example, from previous experiences of the phenomena as a mathematics educator, the researcher bracketed his own experience and knowledge concerning the phenomena under study in order to understand the participants’ experiences entirely by staying away from prejudgment results. In other words, the researcher bracketed his own views about real life examples on Teaching TG and relied on statements supplied by participants.
In phenomenological reduction, the task is to describe individual experiences through textural language. In order to describe the general features of the phenomenon, elements that are not directly within conscious experience were left out (Yuksel & Yildirim, 2015). This was achieved by eliminating overlapping, repetitive, and vague expressions i.e. cleaning the raw data. In this study, there was need to clean the participants’ interview on responses which were not directly linked to the focus of the study.
Imaginative variation is a phenomenological analysis process that follows phenomenological reduction and depends purely on researchers’ imagination rather than empirical data (Yuksel & Yildirim, 2015). The aim was to arrive at structural descriptions of an experience, the underlying and precipitating factors that account for what is being experienced; in other words the “how” that speaks to conditions that illuminate the “what” of experience” (Moustakas, 1994: 85).
Moustakas (1994) defined all research participants as co-researchers because the essence of the phenomena is derived from participants’ perceptions and experiences, regardless of the interpretation of the researcher. The participants’ narratives of experiences provide the meaning of the phenomena. It is the role of the researchers to create the textural, structural, and textural-structural narratives without including their subjectivity (Yuksel & Yildirim, 2015). This means the transcendental analysis requires no interpretation by the researchers.
Structure of the research process
Though qualitative studies are not generalised in the traditional sense, some have redeeming qualities that set them above the requirement (Myers, 2000). According to Yin (2013), qualitative research findings can be transferred to similar contexts. Analytic data cannot be generalised to some defined population that has been sampled, but to a theory of the phenomenon being studied, a theory that may have much wider applicability than the particular phenomenon studied. In this study it resembles experiments in the physical sciences, which make no claim to statistical representativeness, but instead assumes that their results contribute to a general theory of the phenomenon (Yin, 2013). Since the study focused on teachers’ use of students’ real-life experiences in the teaching and learning of geometric transformations it assumes that failure by teachers in using learner experiences in teaching is detrimental to their understanding.
The following is a basic model which was used in the total research process.
The above model was used in an attempt to provide links between components of the research process. For example, theoretical and conceptual frameworks were understood and used in the context of the study’s research questions as well as the goals of the study. Thus, every component was influenced by and influenced at least two other components.
POPULATION AND SAMPLING TECHNIQUES
Selection of the schools and participants was done with a number of considerations. In the following section an account of the research site and the selection of participants is given.
The research site
According to Creswell (2012) a study population refers to a complete group of entities that share a set of characteristics that are similar. The population of this study constituted secondary schools in Mberengwa district. Mberengwa district is one among 10 districts in the midlands province of Zimbabwe. There are 9 provinces in Zimbabwe, giving an estimate of 72 districts in the country. The study purposively selected Mberengwa district schools mainly because of the schools diversity. Basically, they are three types of secondary schools in the district; mission owned schools, government owned schools and council run schools. Three schools were selected purposively, however which the choice resembling the schools diversity. Thus, there are different factors that influence learning in these schools in a significant way, such as students’ home and social life, resources available to the school, and the type of community in which it is situated. The schools were assigned arbitrary names for anonymity; School A, School B and School C.
School A is a co-educational Mission boarding high school of the Evangelical Lutheran Church in Zimbabwe. It is located some 24 km away from a residential township area in the rural Zimbabwe. The school was established a very long time ago. It enrols both boys and girls Forms 1 up to 6. Forms 1 up to 4 have about 4 classes per Form level which are not streamed according to ability. Since it is a mission school the church is responsible for financing all its operations. The school has an average enrolment of about 900 students.
There are four Mathematics teachers only two of whom are professionally qualified. School has boarding facilities that house about 500 boys and girls, with the remainder as ‘day scholars’. The school has a computer laboratory and three separate science laboratories. The computer laboratory has about 20 functional computers. The average number of students is (45) per mathematics class which is tolerable. School A is comparatively well resourced with white boards, fairly well-equipped science laboratories for biology, chemistry and physics practicals. Although School A is a fee-paying school, parents generally can afford the fees. In School A students have their own permanent classes and the teachers move to teach the students during each change-of-lesson time.
School B, is located in a formal residential township. The school was established just after Zimbabwe gained independence in 1980.The school has an average enrolment of about 700 learners in forms 1 – 6. Forms 1 – 4 have each, 3 classes which are not streamed according to ability. Despite its large student population, the school has only 3 mathematics teachers all not qualified (see Table 3.1 for the teachers’ general demographic data).
The Mathematics classes are fairly large, with about 55 learners, typically of many government run schools in Zimbabwe. As with all Government schools students are expected to pay tuition fees and levies. School B is relatively well resourced as it has a computer laboratory with 10 computers donated by the Honourable president of the country, and one science laboratory for the lower classes meant for practicals in integrated science only.
School C, is a ‘day’ secondary school which however is located in a very remote area of the district. It is a school which is located in an area where most parents struggle to raise fees to send their children to school. The school is in a location often hard hit with drought. An average parent in the area is a peasant farmer where proceeds of their sale of agriculture produce would cater for all family expenses.
It enrols both boys and girls Forms 1 – 4 with an average enrolment of 300 students. Each form level has about 2 classes which are not streamed according to ability. The school is classified as a council school and it depends on students’ fees on all its operations. The school is under-resourced with however only 2 qualified maths teachers (see table 3.1 for the teachers’ general demographic data).
The researcher chose mainly purposive sampling (Groenewald, 2004; McMillan & Schumacher, 2006; Teddlie & Yu, 2007) in selecting the research participants. According to Welman and Kruger (2008) purposive sampling is the most important kind of non-probability sampling, to identify the primary participants. Purposive sampling was used to select the mathematics teachers (see Table 3.2 below). According to Richards and Morse (2007) qualitative researchers seek valid representation when they employ non-random sampling techniques such as purposive sampling where participants are chosen based on certain characteristics.
The basic criterion for selection was to look for a mathematics teacher who at that time was teaching an ordinary level class. A sample of participants was selected based on the nature of the research, looking for those who “have had experiences relating to the phenomenon …” of teaching transformation geometry (Teddlie & Yu, 2007). However, in all three schools only one teacher was teaching the ordinary level mathematics classes, so that a second teacher was then selected based on experience in having taught the topic of TG before. Thus a total of six teachers participated in this study.
However, simple random sampling technique was used in selecting students in the ordinary level stream who took the test (see table 3.2 below). Two lessons were observed each for three teachers of mathematics, one on a unit of isometric transformations (translation, reflection or rotation) and the other on non-isometric transformation (shear, stretch or enlargement). The reason being, the researcher wanted to have a feel of the teaching and learning experiences for both types of transformations. However selection of which lessons to observe was somehow a random process so that the flow of lessons at the different schools is not interrupted.
Participation in the study was on voluntary basis and the participants would end their participation in the study at any time without risk or harm. All six teachers participated in the study until it ended. Table 3.1 below shows a summary of the six Mathematics teachers’ demographic data.
The distribution of teachers by gender is biased, showing that all six were male teachers. Of the six four have a teaching qualification. Whilst school A and B have degreed teachers for Mathematics, one of the two at each school has no teaching qualification. According to Shulman (1986) qualified teachers receive training in pedagogical content knowledge necessary to provide a bridge between the subject matter and the knowledge of teaching. This means teacher A2 and teacher B2 are likely not to provide such a bridge in their teaching of concepts in mathematics.
Teachers at School C, whilst they hold a diploma qualification in teaching their long teaching experience might be compromised by the absence of in-service teacher professional development.
CHAPTER ONE THE RESEARCH PROBLEM AND ITS CONTEXT
1.1 NZIRAMASANGA (1999) PRESIDENTIAL COMMISSION OF INQUIRY INTO MATHEMATICS EDUCATION IN ZIMBABWE
1.2 MOTIVATION FOR THE STUDY
1.3 BACKGROUND TO THE STUDY
1.4 STATEMENT OF THE PROBLEM
1.5 EXPLANATORY FRAMEWORK
1.6 RESEARCH QUESTIONS
1.7 SIGNIFICANCE OF THE STUDY
1.8 OBJECTIVES OF THE STUDY
1.9 DELIMITATIONS OF THE STUDY
1.10 RESEARCH DESIGN AND METHODOLOGY
1.11 DEFINITION OF KEY WORDS
1.12 OUTLINE OF THE STUDY
CHAPTER TWO THEORETICAL FRAMEWORK AND REVIEW OF RELATED LITERATURE
2.1 THEORETICAL FRAMEWORK
2.2 CONCEPTUAL FRAMEWORK
2.3 THE CONCEPT OF TRANSFORMATION GEOMETRY
2.4 A HISTORICAL PERSPECTIVE OF THE STRUCTURE OF EUCLIDEAN GEOMETRY
2.5 CREATING SIGNIFICANT LEARNING EXPERIENCES
2.6 RESEARCH FINDINGS ON RME-BASED TEACHING AND LEARNING
2.7 CONSTRUCTIVIST THEORY IN THE TEACHING OF TRANSFORMATIONAL GEOMETRY
2.8 RME PRINCIPLES FOR TASK DESIGN IN TRANSFORMATION GEOMETRY
2.9 THE CONTEXT OF THE STUDY AND TRANSFORMATION GEOMETRY CURRICULUM
2.10 IMPORTANCE OF TRANSFORMATION GEOMETRY IN SCHOOL MATHEMATICS
2.11 THE MATHEMATICS TEXTBOOK
2.12 DIFFICULTIES EXPERIENCED BY LEARNERS WITH TRANSFORMATIONAL GEOMETRY CONCEPTS
2.13 CHAPTER SUMMARY
CHAPTER THREE RESEARCH DESIGN AND METHODOLOGY
3.1 RESEARCH PARADIGM
3.2 METHODOLOGICAL DESIGN
3.3 POPULATION AND SAMPLING TECHNIQUES
3.5 ETHICS AND NEGOTIATING ACCESS
3.6 DATA COLLECTION PROCEDURES
3.7 PHENOMENOLOGICAL DATA ANALYSIS AND REPRESENTATION
3.8 The steps of data analysis
3.9 VALIDITY CONSIDERATIONS
3.11 CHAPTER SUMMARY
CHAPTER FOUR DATA PRESENTATION, ANALYSIS AND DISCUSSION
4.1 RESEARCH QUESTION 1:
4.2 Research Question 2:
4.3 RESEARCH QUESTION 3:
4.4 Research Question 4
4.5 CONCLUSION OF THE CHAPTER
CHAPTER FIVE SUMMARY OF THE STUDY, RECOMMENDATIONS AND CONCLUSION
5.1 Summary of the findings in this study
5.3 RECOMMENDATIONS OF THE STUDY
5.4 LIMITATIONS OF THE STUDY
5.6 FINAL WORD
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